Our manifolds are finite dimensional, Hausdorff and second countable. A smooth manifold is a space with charts whose transition functions are smooth. Its tangent vectors are coordinate velocities, identified by the derivatives of chart transitions. A Lie group is a smooth manifold whose multiplication and inversion are smooth. These are definitions; no subgroup or quotient theorem is included in them.
The support of a function is the closure of the set on which it is nonzero. A compact exhaustion is a sequence of compact sets \\(A_n\\) with \\(A_n\\subset\\operatorname{int}A_{n+1}\\) and \\(\\bigcup_{n\\geq0}A_n=M\\). We write \\(U(a,r)\\) for an open Euclidean ball and \\(B(a,r)\\) for its closed ball.
Lemma 0.5 (a flat smooth cutoff factor). The function \\(\\eta(t)=e^{-1/t^2}\\) for \\(t>0\\), extended by zero for \\(t\\leq0\\), is smooth and all its derivatives vanish at zero.
Lemma 3.B (subordinate atlas). Suppose the topology of the manifold \\(M\\) has a countable basis, and let \\((W_i)_{i\\in I}\\) be an open cover of \\(M\\). There is a countable family of compatible charts \\(\\alpha_j:U_j\\to V_j\\subset\\mathbb R^d\\), \\(j\\in J\\), with radii \\(0<\\delta_j<\\epsilon_j\\) and \\(B(0,\\epsilon_j)\\subset V_j\\), such that
\nDefinition 3.C (partition of unity). On a topological space \\(X\\), a family \\((h_j)_{j\\in J}\\) of functions \\(h_j:X\\to[0,1]\\) is a partition of unity if every point of \\(X\\) has a neighbourhood on which all but finitely many \\(h_j\\) vanish identically, and \\(\\sum_{j\\in J}h_j(x)=1\\) for every \\(x\\in X\\). The first condition makes each of these sums finite. The partition is called continuous, differentiable or smooth when every \\(h_j\\) is.
Definition 3.D (subordinate partition). For an open cover \\((W_i)_{i\\in I}\\) of a topological space \\(X\\), a partition of unity \\((h_j)_{j\\in J}\\) on \\(X\\) is subordinate to the cover if for every \\(j\\in J\\) the support of \\(h_j\\) lies in some member \\(W_{i(j)}\\).
Theorem 3.E (partition of unity). Suppose the topology of the differentiable manifold \\(M\\) has a countable basis. Every open cover of \\(M\\) has a subordinate partition of unity consisting of continuously differentiable functions, whose supports are compact and lie in coordinate domains contained in members of the cover. If \\(M\\) is smooth, the partition can be chosen smooth.
Theorem 3.1 (smooth partition and cutoff). Every open cover of a smooth manifold admits a locally finite smooth partition of unity subordinate to it. Its supports can be chosen compact and contained in coordinate neighbourhoods lying in members of the cover. If \\(K\\) is closed and \\(U\\) is an open neighbourhood of \\(K\\), there is a smooth function \\(\\chi:M\\to[0,1]\\) equal to one on a neighbourhood of \\(K\\), with support contained in \\(U\\).
", "availability": "statement_and_proof_present", "proof_links": [ { "reader": "docs/courses/AG-GS/prerequisites/curated/courses/DG-FND/local-tools-for-bundles-and-transport.html#0-analytic-and-linear-foundations", "label": "Lemma 0.5: the flat smooth cutoff factor and its proof" } ], "source_credit_html": "Statement text: GPT-6 Astra (OpenAI), Local tools for bundles and transport (original text CC0). The Section 3 construction follows Holger Brenner and Wikiversity contributors, Lecture 22 of Differentialgeometrie (Osnabrück 2023), revision 1052940.
" }, { "id": "compact-exhaustion", "title": "Compact exhaustion of a manifold", "lesson_id": "local-tools-for-bundles-and-transport", "course_id": "DG-FND", "lesson_label": "Local tools for bundles and transport", "reader": "docs/courses/AG-GS/prerequisites/curated/courses/DG-FND/local-tools-for-bundles-and-transport.html#3-smooth-weights-with-controlled-support", "reader_label": "Statements and proofs", "source": "docs/courses/AG-GS/prerequisites/curated/courses/DG-FND/src/local-tools-for-bundles-and-transport.md", "source_sha256": "bbc71638ea4219576097b63b068a8222dc1e2699f007f45c8fab45302390a2aa", "reader_sha256": "087b126c5be5587ab6b787c8616668c70098dab76382fdaf94d88726ff25dd13", "statement_html": "Our manifolds are finite dimensional, Hausdorff and second countable. A smooth manifold is a space with charts whose transition functions are smooth. Its tangent vectors are coordinate velocities, identified by the derivatives of chart transitions. A Lie group is a smooth manifold whose multiplication and inversion are smooth. These are definitions; no subgroup or quotient theorem is included in them.
The support of a function is the closure of the set on which it is nonzero. A compact exhaustion is a sequence of compact sets \\(A_n\\) with \\(A_n\\subset\\operatorname{int}A_{n+1}\\) and \\(\\bigcup_{n\\geq0}A_n=M\\). We write \\(U(a,r)\\) for an open Euclidean ball and \\(B(a,r)\\) for its closed ball.
Lemma 3.A (compact exhaustion). Every manifold \\(M\\) whose topology has a countable basis admits a compact exhaustion.
", "availability": "statement_and_proof_present", "source_credit_html": "Statement text: GPT-6 Astra (OpenAI), Local tools for bundles and transport (original text CC0). The Section 3 construction follows Holger Brenner and Wikiversity contributors, Lecture 22 of Differentialgeometrie (Osnabrück 2023), revision 1052940.
" }, { "id": "AG-QC-01.theorem-5-1", "title": "Two open sets and the connecting map", "lesson_id": "AG-QC-01", "reader": "docs/courses/AG-QC/cohomology-of-sheaves-on-ringed-spaces.html#5-two-open-sets-and-the-connecting-map", "source": "docs/courses/AG-QC/src/cohomology-of-sheaves-on-ringed-spaces.md", "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977", "reader_sha256": "ebe9407afa807fc50f21b4ad20ead65531a1765c043efd6ca03027bdc181901e", "statement_html": "Theorem 5.1 (Mayer–Vietoris). If \\(X=U\\cup V\\) is an open cover, there is a natural exact sequence\n\\[\n\\begin{aligned}\n0&\\to H^0(X,\\mathcal F)\n\\to H^0(U,\\mathcal F)\\oplus H^0(V,\\mathcal F)\n\\to H^0(U\\cap V,\\mathcal F)\\\\\n&\\to H^1(X,\\mathcal F)\n\\to H^1(U,\\mathcal F)\\oplus H^1(V,\\mathcal F)\n\\to H^1(U\\cap V,\\mathcal F)\\to\\cdots.\n\\end{aligned}\n\\]\nThe map from the two opens to their intersection is restriction from \\(U\\) minus restriction from \\(V\\).
", "availability": "statement_and_proof_present" }, { "id": "AG-QC-02.theorem-2-1", "title": "A smaller complex without repeated indices", "lesson_id": "AG-QC-02", "reader": "docs/courses/AG-QC/cech-cohomology.html#2-a-smaller-complex-without-repeated-indices", "source": "docs/courses/AG-QC/src/cech-cohomology.md", "source_sha256": "63988581823cbb25c6f3678c41c9c743add99fe8753175922fb059e5972b5871", "reader_sha256": "bb671a20137ba6d2e09eb5010d65c5ccfb7238ac469be119796cd49ccf3a7c2e", "statement_html": "Theorem 2.1. The ordered and alternating complexes are isomorphic, and their inclusion into the full complex is a homotopy equivalence. No division by a factorial is involved.
", "availability": "statement_and_proof_present" }, { "id": "AG-QC-02.theorem-3-2", "title": "Comparing overlaps with derived cohomology", "lesson_id": "AG-QC-02", "reader": "docs/courses/AG-QC/cech-cohomology.html#3-comparing-overlaps-with-derived-cohomology", "source": "docs/courses/AG-QC/src/cech-cohomology.md", "source_sha256": "63988581823cbb25c6f3678c41c9c743add99fe8753175922fb059e5972b5871", "reader_sha256": "bb671a20137ba6d2e09eb5010d65c5ccfb7238ac469be119796cd49ccf3a7c2e", "statement_html": "Theorem 3.2 (acyclic covers). If \\(H^q(U_{i_0\\ldots i_p},\\mathcal F)=0\\) for all \\(q>0\\) and every nonempty finite intersection, then the natural comparison is an isomorphism\n\\[\n\\check H^p(\\mathcal U,\\mathcal F)\\cong H^p(X,\\mathcal F)\n\\]\nfor every \\(p\\).
", "availability": "statement_and_proof_present" }, { "id": "AG-QC-01.long-exact-cohomology-sequence", "title": "The obstruction that cohomology records", "lesson_id": "AG-QC-01", "reader": "docs/courses/AG-QC/cohomology-of-sheaves-on-ringed-spaces.html#1-the-obstruction-that-cohomology-records", "source": "docs/courses/AG-QC/src/cohomology-of-sheaves-on-ringed-spaces.md", "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977", "reader_sha256": "ebe9407afa807fc50f21b4ad20ead65531a1765c043efd6ca03027bdc181901e", "statement_html": null, "availability": "statement_and_derived_functor_route_present" }, { "id": "AG-QC-02.theorem-5-1", "title": "Torsors are first cohomology classes", "lesson_id": "AG-QC-02", "reader": "docs/courses/AG-QC/cech-cohomology.html#5-torsors-are-first-cohomology-classes", "source": "docs/courses/AG-QC/src/cech-cohomology.md", "source_sha256": "63988581823cbb25c6f3678c41c9c743add99fe8753175922fb059e5972b5871", "reader_sha256": "bb671a20137ba6d2e09eb5010d65c5ccfb7238ac469be119796cd49ccf3a7c2e", "statement_html": "Theorem 5.1. Isomorphism classes of \\(\\mathcal F\\)-torsors are naturally classified by \\(H^1(X,\\mathcal F)\\). The comparison\n\\(\\check H^1(\\mathcal U,\\mathcal F)\\to H^1(X,\\mathcal F)\\)\nis injective, and its image consists exactly of classes of torsors trivial on every \\(U_i\\).
", "availability": "statement_and_proof_present" }, { "id": "AG-QC-01.theorem-3-1", "title": "Coefficients and higher direct images", "lesson_id": "AG-QC-01", "reader": "docs/courses/AG-QC/cohomology-of-sheaves-on-ringed-spaces.html#3-coefficients-and-higher-direct-images", "source": "docs/courses/AG-QC/src/cohomology-of-sheaves-on-ringed-spaces.md", "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977", "reader_sha256": "ebe9407afa807fc50f21b4ad20ead65531a1765c043efd6ca03027bdc181901e", "statement_html": "Theorem 3.1 (independence of the coefficient category). The underlying groups of \\(H^q(U,\\mathcal F)\\), computed in modules over \\(\\mathcal O_X\\), are naturally the cohomology groups of the underlying abelian sheaf.
", "availability": "statement_and_proof_present" }, { "id": "AG-QC-01.theorem-2-3+corollary-2-4", "title": "Extending sections removes the obstruction", "lesson_id": "AG-QC-01", "reader": "docs/courses/AG-QC/cohomology-of-sheaves-on-ringed-spaces.html#2-extending-sections-removes-the-obstruction", "source": "docs/courses/AG-QC/src/cohomology-of-sheaves-on-ringed-spaces.md", "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977", "reader_sha256": "ebe9407afa807fc50f21b4ad20ead65531a1765c043efd6ca03027bdc181901e", "statement_html": "Theorem 2.3 (flasque acyclicity). For a flasque sheaf of modules \\(\\mathcal F\\),\n\\[\nH^q(U,\\mathcal F)=0\\qquad(q>0)\n\\]\non every open \\(U\\subset X\\).
\nCorollary 2.4 (flasque resolutions). An exact resolution \\(0\\to\\mathcal F\\to\\mathcal L^0\\to\\mathcal L^1\\to\\cdots\\) with all \\(\\mathcal L^r\\) flasque computes cohomology on every open by taking sections.
", "availability": "statement_and_proof_present" }, { "id": "distribution-local-calculus", "title": "Local differential rules for distributions", "lesson_id": "local-data-and-compatible-products", "course_id": "AN-01", "lesson_label": "Local data and compatible products", "reader": "docs/courses/AN-01/reader/AN01-U009.html#the-smooth-locus-and-local-operations", "reader_label": "Statement and argument", "source": "docs/courses/AN-01/src/local-data-and-compatible-products.md", "source_sha256": "3afcebad6619d171988696dfe2138a5b36d01fb73bf6bf9a70b1e976c98e8e2a", "reader_sha256": "91b3238dd59b1d6499a84d307c736c478cf2175b5027cd952937889b6ec32e79", "statement_html": "Theorem 3.1 (support-relative test extension). Let \\(u\\in\\mathcal D'(X)\\) and let \\(F\\subset X\\) be relatively closed, with \\(\\operatorname{supp}u\\subset F\\). On the vector space
\n\\[\n\\mathcal E_F(X)=\n\\{f\\in C^\\infty(X):\nF\\cap\\operatorname{supp}f\\text{ is compact in }X\\},\n\\tag{3.1}\n\\]\nthere is a unique linear extension \\(\\widetilde u\\) of \\(u\\) that vanishes whenever \\(\\operatorname{supp}f\\cap F=\\varnothing\\). If \\(\\chi\\in C_c^\\infty(X)\\) equals one near \\(F\\cap\\operatorname{supp}f\\), its formula is
\n\\[\n\\widetilde u(f)=u(\\chi f).\n\\tag{3.2}\n\\]\nWhen \\(u\\) has order at most \\(k\\), the same construction extends its canonical \\(C_c^k\\) pairing to the space defined by (3.1) with \\(C^k(X)\\) in place of \\(C^\\infty(X)\\).
Define the singular support \\(\\operatorname{sing\\,supp}u\\) as the complement of the union of open sets where \\(u\\) is represented by a smooth function.
For \\(a\\in C^\\infty(X)\\), define
\n\\[\n(\\partial_j u)(\\phi)=-u(\\partial_j\\phi),\n\\qquad\n(au)(\\phi)=u(a\\phi).\n\\tag{4.2}\n\\]Proposition 4.1 (local differential rules). The operations (4.2) give distributions, commute with restriction, and are continuous in both the weak and strong dual topologies for fixed \\(a\\). They satisfy
\n\\[\n\\begin{gathered}\n\\operatorname{supp}(\\partial_j u)\\subset\\operatorname{supp}u,\n\\qquad\n\\operatorname{supp}(au)\\subset\\operatorname{supp}a\\cap\\operatorname{supp}u,\\\\\n\\operatorname{sing\\,supp}(\\partial_j u)\\subset\\operatorname{sing\\,supp}u,\n\\qquad\n\\operatorname{sing\\,supp}(au)\\subset\\operatorname{supp}a\\cap\\operatorname{sing\\,supp}u,\\\\\n\\partial^\\alpha(au)=\n\\sum_{\\beta\\le\\alpha}\\binom{\\alpha}{\\beta}\n(\\partial^\\beta a)\\partial^{\\alpha-\\beta}u.\n\\end{gathered}\n\\tag{4.3}\n\\]\nPartial derivatives commute. Both formulas in (4.2) also hold for every \\(f\\in\\mathcal E_{\\operatorname{supp}u}(X)\\), using the extended pairings on their left sides.
\nIf \\(a\\in C^k(X)\\) and \\(u\\) has order at most \\(k\\), the second formula still defines \\(au\\), by the \\(C_c^k\\) pairing. It has order at most \\(k\\), with support contained in the same support intersection. The singular-support assertions require a smooth multiplier.
", "availability": "statement_and_proof_present", "proof_links": [ { "reader": "docs/courses/AN-01/reader/AN01-U009.html#tests-that-are-compact-only-on-the-support", "label": "Theorem 3.1: support-relative pairing and proof" } ] }, { "id": "distributional-constants-on-an-interval", "title": "Zero distributional derivative on an interval", "lesson_id": "weak-equations-and-classical-functions", "course_id": "AN-01", "lesson_label": "Weak equations and classical functions", "reader": "docs/courses/AN-01/reader/AN01-U010.html#a-zero-derivative-determines-a-constant", "reader_label": "Statements and proofs", "source": "docs/courses/AN-01/src/weak-equations-and-classical-functions.md", "source_sha256": "efdb4a90d8e0c9d009fb80c37b6e7b36721928efd34a0669647db16ee16e48a1", "reader_sha256": "f8ca316fec608568a7ab0ed72286115c2a835c1a9b9664fcbfdaaf45198aa2f0", "statement_html": "Theorem 1.1 (distributional constants). If \\(I\\subset\\mathbb R\\) is a nonempty open interval and \\(u\\in\\mathcal D'(I)\\) satisfies \\(u'=0\\), there is a unique scalar \\(c\\) such that
\n\\[\nu(\\phi)=c\\int_I\\phi(t)\\,dt.\n\\tag{1.1}\n\\]\nThe interval may be unbounded. On a disconnected open set the constant can differ between its interval components.
", "availability": "statement_and_proof_present" }, { "id": "complex-analytic-spaces-and-coherent-sheaves.frechet-spaces-of-sections-and-schwartzs-theorem.theorem-1-1", "title": "The open mapping theorem for Fréchet spaces", "lesson_id": "frechet-spaces-of-sections-and-schwartzs-theorem", "course_id": "complex-analytic-spaces-and-coherent-sheaves", "lesson_label": "Fréchet spaces of sections and Schwartz's theorem", "reader": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/frechet-spaces-of-sections-and-schwartzs-theorem.html#1-fréchet-spaces", "reader_label": "Statement and proof", "source": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/src/frechet-spaces-of-sections-and-schwartzs-theorem.md", "source_sha256": "96d01ef852dff592655f54430ff79ad1b43799b970d41c1e7dc4a8e988ba8c3a", "reader_sha256": "b45eafbe62f114626b95f199a030578a8d842ce1a3731aa1b1bcd6536893e3da", "statement_html": "Theorem 1.1 (open mapping theorem). A continuous linear surjection \\(A:E\\to F\\) between Fréchet spaces is open. A continuous linear bijection between Fréchet spaces is a topological isomorphism.
", "availability": "statement_and_proof_present" }, { "id": "complex-analytic-spaces-and-coherent-sheaves.frechet-spaces-of-sections-and-schwartzs-theorem.theorem-2-2+remark-2-3", "title": "Compact cohomology and quotient topologies", "lesson_id": "frechet-spaces-of-sections-and-schwartzs-theorem", "course_id": "complex-analytic-spaces-and-coherent-sheaves", "lesson_label": "Fréchet spaces of sections and Schwartz's theorem", "reader": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/frechet-spaces-of-sections-and-schwartzs-theorem.html#2-schwartz-s-theorem", "reader_label": "Full statements and arguments", "source": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/src/frechet-spaces-of-sections-and-schwartzs-theorem.md", "source_sha256": "96d01ef852dff592655f54430ff79ad1b43799b970d41c1e7dc4a8e988ba8c3a", "reader_sha256": "b45eafbe62f114626b95f199a030578a8d842ce1a3731aa1b1bcd6536893e3da", "statement_html": "Theorem 2.2 (finiteness criterion). Let \\(\\rho:(E^\\bullet,d)\\to(F^\\bullet,\\delta)\\) be a morphism of complexes of Fréchet spaces with continuous differentials. If \\(\\rho^q\\) is compact and \\(H^q(\\rho):H^q(E^\\bullet)\\to H^q(F^\\bullet)\\) is surjective, then \\(H^q(F^\\bullet)\\) is finite-dimensional and \\(\\delta(F^{q-1})\\) is closed in \\(F^q\\).
\nProof. Consider \\(g,h:Z^q(E^\\bullet)\\oplus F^{q-1}\\to Z^q(F^\\bullet)\\), \\(g(x,y)=\\rho^q(x)+\\delta y\\), \\(h(x,y)=-\\rho^q(x)\\), between Fréchet spaces (cocycle spaces are closed). Surjectivity of \\(H^q(\\rho)\\) means that \\(g\\) is surjective, and \\(h\\) is compact. By Theorem 2.1, \\(g+h=(x,y)\\mapsto\\delta y\\) is a quasi-epimorphism, so \\(\\delta(F^{q-1})\\) is closed and of finite codimension in \\(Z^q(F^\\bullet)\\). \\(\\square\\)
\nRemark 2.3. In the situation of Theorem 2.2 without compactness, if \\(H^q(\\rho)\\) is surjective then the map \\(g\\) is open by Theorem 1.1, so \\(H^q(\\rho)\\) is open for the quotient topologies; if \\(H^q(\\rho)\\) is bijective it is a homeomorphism.
", "availability": "statement_and_proof_present" }, { "id": "complex-analytic-spaces-and-coherent-sheaves.frechet-spaces-of-sections-and-schwartzs-theorem.proposition-5-1+proposition-5-2", "title": "The topology and compact restriction of coherent sections", "lesson_id": "frechet-spaces-of-sections-and-schwartzs-theorem", "course_id": "complex-analytic-spaces-and-coherent-sheaves", "lesson_label": "Fréchet spaces of sections and Schwartz's theorem", "reader": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/frechet-spaces-of-sections-and-schwartzs-theorem.html#5-the-topology-on-sections-of-coherent-sheaves", "reader_label": "Full statements and arguments", "source": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/src/frechet-spaces-of-sections-and-schwartzs-theorem.md", "source_sha256": "96d01ef852dff592655f54430ff79ad1b43799b970d41c1e7dc4a8e988ba8c3a", "reader_sha256": "b45eafbe62f114626b95f199a030578a8d842ce1a3731aa1b1bcd6536893e3da", "statement_html": "Let \\(X\\) be a complex space and \\(\\mathcal S\\) a coherent \\(\\mathcal O_X\\)-module. Call an open set \\(U\\subset X\\) standard if it is of the form \\(U=A\\cap V\\), where \\(A\\subset\\Omega\\subset\\mathbf C^N\\) is a distinguished patch for \\(\\mathcal S\\) (Definition 4.2 of the previous lesson) with resolution beginning \\(\\mathcal O_\\Omega^{p_0}\\to i_*\\mathcal S\\to0\\), and \\(V\\subset\\Omega\\) is a Stein domain. By Proposition 4.3 of the previous lesson, the sequence
\n\\[\n0\\longrightarrow\\mathcal Z_0(V)\\longrightarrow\\mathcal O(V)^{p_0}\\longrightarrow\\mathcal S(U)\\longrightarrow0\n\\tag{5.1}\n\\]\nis exact, and \\(\\mathcal Z_0(V)\\) is closed by Section 4. Give \\(\\mathcal S(U)\\) the quotient Fréchet topology.
\nProposition 5.1. This topology does not depend on the choices. For an arbitrary open set \\(W\\subset X\\), the topology on \\(\\mathcal S(W)\\) induced by the injection \\(\\mathcal S(W)\\to\\prod_\\alpha\\mathcal S(U_\\alpha)\\), for any countable covering of \\(W\\) by standard open sets, is a Fréchet topology independent of the covering, and it agrees with the previous one when \\(W\\) is standard.
\nProposition 5.2. Let \\(\\mathcal S\\) be coherent on \\(X\\).
\nRegistered statement. Every countable product of real or complex Frechet spaces is Frechet for the product topology; finite and empty families included.
", "availability": "statement_and_proof_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_pointer": "/exact_product_finite_dimensional_inputs_20261013/locally_proved_result" } }, { "id": "complex-analytic-spaces-and-coherent-sheaves.frechet-spaces-of-sections-and-schwartzs-theorem.proposition-3-1", "title": "The abstract Mittag-Leffler theorem", "lesson_id": "frechet-spaces-of-sections-and-schwartzs-theorem", "course_id": "complex-analytic-spaces-and-coherent-sheaves", "lesson_label": "Fréchet spaces of sections and Schwartz's theorem", "reader": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/frechet-spaces-of-sections-and-schwartzs-theorem.html#3-an-abstract-mittag-leffler-theorem", "reader_label": "Statement and proof", "source": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/src/frechet-spaces-of-sections-and-schwartzs-theorem.md", "source_sha256": "96d01ef852dff592655f54430ff79ad1b43799b970d41c1e7dc4a8e988ba8c3a", "reader_sha256": "b45eafbe62f114626b95f199a030578a8d842ce1a3731aa1b1bcd6536893e3da", "statement_html": "Proposition 3.1. Let \\((E^\\bullet_\\nu)_{\\nu\\in\\mathbf N}\\) be complexes of Fréchet spaces with continuous differentials and continuous morphisms \\(E^\\bullet_{\\nu+1}\\to E^\\bullet_\\nu\\) with dense image in each degree, and let \\(E^\\bullet=\\varprojlim E^\\bullet_\\nu\\).
\nHere density in cohomology refers to the quotient topology on cocycles modulo coboundaries.
", "availability": "statement_and_proof_present" }, { "id": "derived-categories-and-sheaf-operations.injective-modules-and-bounded-below-derived-functors.theorem-2-3+lemma-1-1", "title": "Injective embeddings, flasqueness and restriction", "lesson_id": "injective-modules-and-bounded-below-derived-functors", "course_id": "derived-categories-and-sheaf-operations", "lesson_label": "Injective modules, flasque sheaves and bounded-below derived functors", "reader": "docs/courses/derived-categories-and-sheaf-operations/injective-modules-and-bounded-below-derived-functors.html#2-functorial-injective-embeddings", "reader_label": "Theorem 2.3: statement and proof", "proof_links": [ { "label": "Lemma 1.1: flasqueness and restriction", "reader": "docs/courses/derived-categories-and-sheaf-operations/injective-modules-and-bounded-below-derived-functors.html#1-extending-maps-and-extending-sections" } ], "source": "docs/courses/derived-categories-and-sheaf-operations/src/injective-modules-and-bounded-below-derived-functors.md", "source_sha256": "b2dc989efbc8d2d4004c8d24cbc3004a9e81da485cd702bb7d293a7f8af5edba", "reader_sha256": "d2e87fa29a80deac8b72c036dffd3d35348aa2d90fb645e14b8ae846f4cc3be3", "statement_html": "Theorem 2.3 (functorial injective embeddings). Let \\(\\mathcal O\\) be any sheaf of rings on \\(X\\). Put\n\\[\nE(F)=\\prod_{x\\in X}x_*\\bigl(I_{\\mathcal O_x}(F_x)\\bigr),\n\\]\nwhere \\(I_{\\mathcal O_x}\\) is the functor (2.1) for the ring \\(\\mathcal O_x\\), and let \\(\\iota_F:F\\to E(F)\\) have \\(x\\)-component the morphism that corresponds under (5.2) of the preceding lesson to \\(\\iota_{F_x}:F_x\\to I_{\\mathcal O_x}(F_x)\\). Then \\(E(F)\\) is injective, \\(\\iota_F\\) is a monomorphism, and \\(E\\) is a functor with \\(\\iota\\) natural. So \\(\\operatorname{Mod}(\\mathcal O)\\) has enough injectives, with functorial injective embeddings.
\nLemma 1.1 (injectives are flasque and restrict to injectives). An injective module sheaf is flasque. Its restriction to any open subset is injective in the module category of that open subset.
", "availability": "statement_and_proof_present" }, { "id": "derived-categories-and-sheaf-operations.injective-modules-and-bounded-below-derived-functors.theorem-4-1", "title": "Right derived functors on bounded-below complexes", "lesson_id": "injective-modules-and-bounded-below-derived-functors", "course_id": "derived-categories-and-sheaf-operations", "lesson_label": "Injective modules, flasque sheaves and bounded-below derived functors", "reader": "docs/courses/derived-categories-and-sheaf-operations/injective-modules-and-bounded-below-derived-functors.html#4-right-derived-functors-on-bounded-below-complexes", "reader_label": "Full statement of Theorem 4.1", "proof_links": [ { "label": "Proof of part (1): constructing an injective resolution", "reader": "docs/courses/derived-categories-and-sheaf-operations/injective-modules-and-bounded-below-derived-functors.html#constructing-the-resolution-one-degree-at-a-time" }, { "label": "Proof of parts (2)–(3): comparison and derived functors", "reader": "docs/courses/derived-categories-and-sheaf-operations/injective-modules-and-bounded-below-derived-functors.html#comparison-triangulation-and-universality" }, { "label": "Proof of parts (4)–(5): acyclic resolutions", "reader": "docs/courses/derived-categories-and-sheaf-operations/injective-modules-and-bounded-below-derived-functors.html#why-acyclic-terms-suffice-in-the-bounded-below-case" } ], "source": "docs/courses/derived-categories-and-sheaf-operations/src/injective-modules-and-bounded-below-derived-functors.md", "source_sha256": "b2dc989efbc8d2d4004c8d24cbc3004a9e81da485cd702bb7d293a7f8af5edba", "reader_sha256": "d2e87fa29a80deac8b72c036dffd3d35348aa2d90fb645e14b8ae846f4cc3be3", "statement_html": "Theorem 4.1. Let \\(\\mathcal A\\) and \\(\\mathcal B\\) be abelian, and assume that every object of \\(\\mathcal A\\) embeds in an injective object. Let \\(\\Phi:\\mathcal A\\to\\mathcal B\\) be additive.
\nRegistered conclusion. Metric and seminorm convergence and Cauchy conditions coincide; equivalent countable families preserve completeness; closed subspaces and finite products are Frechet.
", "availability": "registered_conclusion_and_argument_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_id": "AN03-Frechet-metric-and-closed-subspaces", "field": "conclusion" } }, { "id": "AN03-Frechet-closed-quotient", "title": "Complete Fréchet quotients", "lesson_id": "banach-foundation-bridges", "course_id": "elliptic-boundary-reduction", "lesson_label": "Banach\nestimates, quotient spaces and compact parameter arguments", "reader": "docs/courses/elliptic-boundary-reduction/banach-foundation-bridges.html#AN03-BFD-FRECHET-QUOTIENT-001", "reader_label": "Statement and argument", "source": "docs/courses/elliptic-boundary-reduction/src/banach-foundation-bridges.md", "source_sha256": "974787be9fa8002995a8758d07a55baa81308480ccf7834e677d43d0c0978659", "reader_sha256": "c68c13536d24453de6e86f64e8bca7b142e150fbd65b130909e03010e2634ba9", "statement_html": "Registered conclusion. E/N with its quotient topology is Hausdorff, locally convex, metrizable and complete; the quotient map is continuous, linear and open.
", "availability": "registered_conclusion_and_argument_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_id": "AN03-Frechet-closed-quotient", "field": "conclusion" } }, { "id": "AN03-Frechet-open-mapping-alternative", "title": "An alternative Fréchet open-mapping proof", "lesson_id": "banach-foundation-bridges", "course_id": "elliptic-boundary-reduction", "lesson_label": "Banach\nestimates, quotient spaces and compact parameter arguments", "reader": "docs/courses/elliptic-boundary-reduction/banach-foundation-bridges.html#AN03-BFD-FRECHET-004", "reader_label": "Statement and argument", "source": "docs/courses/elliptic-boundary-reduction/src/banach-foundation-bridges.md", "source_sha256": "974787be9fa8002995a8758d07a55baa81308480ccf7834e677d43d0c0978659", "reader_sha256": "c68c13536d24453de6e86f64e8bca7b142e150fbd65b130909e03010e2634ba9", "statement_html": "Registered conclusion. The map is open; in the bijective case the inverse is continuous.
", "availability": "registered_conclusion_and_argument_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_id": "AN03-Frechet-open-mapping-alternative", "field": "conclusion" }, "related_results": [ { "id": "complex-analytic-spaces-and-coherent-sheaves.frechet-spaces-of-sections-and-schwartzs-theorem.theorem-1-1", "label": "The local open-mapping theorem and its proof" } ] }, { "id": "completed-Lebesgue-measure", "title": "Completed Euclidean Lebesgue measure", "lesson_id": "banach-foundation-bridges", "course_id": "elliptic-boundary-reduction", "lesson_label": "Banach\nestimates, quotient spaces and compact parameter arguments", "reader": "docs/courses/elliptic-boundary-reduction/banach-foundation-bridges.html#constructing-the-measure-without-importing-a-convergence-theorem", "reader_label": "Statement and argument", "source": "docs/courses/elliptic-boundary-reduction/src/banach-foundation-bridges.md", "source_sha256": "974787be9fa8002995a8758d07a55baa81308480ccf7834e677d43d0c0978659", "reader_sha256": "c68c13536d24453de6e86f64e8bca7b142e150fbd65b130909e03010e2634ba9", "statement_html": "Registered conclusion and application. Volume, countable additivity, Borel measurability and null completion; used for the scalar L2 spaces.
", "availability": "registered_application_and_argument_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_id": "completed-Lebesgue-measure", "field": "conclusion_and_consumer_match" } }, { "id": "dominated-convergence-product-integration", "title": "Convergence and Euclidean product integration", "lesson_id": "banach-foundation-bridges", "course_id": "elliptic-boundary-reduction", "lesson_label": "Banach\nestimates, quotient spaces and compact parameter arguments", "reader": "docs/courses/elliptic-boundary-reduction/banach-foundation-bridges.html#convergence-product-integration-and-the-full-linear-jacobian", "reader_label": "Statement and argument", "source": "docs/courses/elliptic-boundary-reduction/src/banach-foundation-bridges.md", "source_sha256": "974787be9fa8002995a8758d07a55baa81308480ccf7834e677d43d0c0978659", "reader_sha256": "c68c13536d24453de6e86f64e8bca7b142e150fbd65b130909e03010e2634ba9", "statement_html": "Registered conclusion and application. Monotone and dominated convergence and Euclidean Tonelli-Fubini; weighted cutoff tails use the integrable function |f|^2 exp(-phi).
", "availability": "registered_application_and_argument_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_id": "dominated-convergence-product-integration", "field": "conclusion_and_consumer_match" } }, { "id": "holder-minkowski-young", "title": "Hölder, Minkowski and Young inequalities", "lesson_id": "banach-foundation-bridges", "course_id": "elliptic-boundary-reduction", "lesson_label": "Banach\nestimates, quotient spaces and compact parameter arguments", "reader": "docs/courses/elliptic-boundary-reduction/banach-foundation-bridges.html#hölder-minkowski-and-all-young-endpoints", "reader_label": "Statement and argument", "source": "docs/courses/elliptic-boundary-reduction/src/banach-foundation-bridges.md", "source_sha256": "974787be9fa8002995a8758d07a55baa81308480ccf7834e677d43d0c0978659", "reader_sha256": "c68c13536d24453de6e86f64e8bca7b142e150fbd65b130909e03010e2634ba9", "statement_html": "Registered conclusion and application. Cauchy-Schwarz for local test pairings and the L1-L2 convolution bound needed for the uniform commutator estimate.
", "availability": "registered_application_and_argument_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_id": "holder-minkowski-young", "field": "conclusion_and_consumer_match" } }, { "id": "Lp-completeness-density", "title": "Lebesgue completeness and finite-exponent density", "lesson_id": "banach-foundation-bridges", "course_id": "elliptic-boundary-reduction", "lesson_label": "Banach\nestimates, quotient spaces and compact parameter arguments", "reader": "docs/courses/elliptic-boundary-reduction/banach-foundation-bridges.html#completeness-and-compact-smooth-density", "reader_label": "Statement and argument", "source": "docs/courses/elliptic-boundary-reduction/src/banach-foundation-bridges.md", "source_sha256": "974787be9fa8002995a8758d07a55baa81308480ccf7834e677d43d0c0978659", "reader_sha256": "c68c13536d24453de6e86f64e8bca7b142e150fbd65b130909e03010e2634ba9", "statement_html": "Registered conclusion and application. Unweighted L2 completeness; multiplication by exp(-phi/2) transfers it isometrically to each continuous-weight form space. No L-infinity translation-density assertion.
", "availability": "registered_application_and_argument_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_id": "Lp-completeness-density", "field": "conclusion_and_consumer_match" } }, { "id": "scaled-mollifier-approximation", "title": "Scaled smooth approximation", "lesson_id": "banach-foundation-bridges", "course_id": "elliptic-boundary-reduction", "lesson_label": "Banach\nestimates, quotient spaces and compact parameter arguments", "reader": "docs/courses/elliptic-boundary-reduction/banach-foundation-bridges.html#the-original-scaled-approximate-identity", "reader_label": "Statement and argument", "source": "docs/courses/elliptic-boundary-reduction/src/banach-foundation-bridges.md", "source_sha256": "974787be9fa8002995a8758d07a55baa81308480ccf7834e677d43d0c0978659", "reader_sha256": "c68c13536d24453de6e86f64e8bca7b142e150fbd65b130909e03010e2634ba9", "statement_html": "Registered conclusion and application. Smooth convolution, its derivative formulas and L2 convergence. A fixed compact upper weight bound gives the weighted convergence used here.
", "availability": "registered_application_and_argument_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_id": "scaled-mollifier-approximation", "field": "conclusion_and_consumer_match" } }, { "id": "weak-derivative-limits", "title": "Weak-derivative limits and local approximation", "lesson_id": "banach-foundation-bridges", "course_id": "elliptic-boundary-reduction", "lesson_label": "Banach\nestimates, quotient spaces and compact parameter arguments", "reader": "docs/courses/elliptic-boundary-reduction/banach-foundation-bridges.html#complete-weak-derivative-limits-and-sobolev-approximation", "reader_label": "Statement and argument", "source": "docs/courses/elliptic-boundary-reduction/src/banach-foundation-bridges.md", "source_sha256": "974787be9fa8002995a8758d07a55baa81308480ccf7834e677d43d0c0978659", "reader_sha256": "c68c13536d24453de6e86f64e8bca7b142e150fbd65b130909e03010e2634ba9", "statement_html": "Registered conclusion and application. Distributional limit continuity, cutoff Leibniz rule and derivative-convolution commutation. Provider convention D=-i partial is converted explicitly.
", "availability": "registered_application_and_argument_present", "statement_origin": { "register": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/providers.json", "record_id": "weak-derivative-limits", "field": "conclusion_and_consumer_match" } }, { "id": "coordinate-ball-cutoff-alternative", "title": "Explicit cutoffs on coordinate balls", "lesson_id": "metric-foundation-bridges", "course_id": "elliptic-boundary-reduction", "lesson_label": "Metric and topological\nfoundations", "reader": "docs/courses/elliptic-boundary-reduction/metric-foundation-bridges.html#the-actual-smooth-cutoff-with-all-endpoint-constants", "reader_label": "Construction and proof", "source": "docs/courses/elliptic-boundary-reduction/src/metric-foundation-bridges.md", "source_sha256": "240f732e420f4378d91a57c07be45ac6259f9b15762e11ef614c4e895203b49c", "reader_sha256": "56b7f8b925d1d6a932464d1e587c4be3fdca6400eef805686d4a176c3887d204", "statement_html": "Keep the function from this lesson:
\n\\[\n \\eta(s)=\\begin{cases}0,&s\\le0,\\\\ E(-1/s),&s>0.\\end{cases}\n \\qquad\n \\chi(s)=\\frac{\\eta(3/2-s)}\n {\\eta(3/2-s)+\\eta(s-1)}.\n\\tag{OC38}\n\\]
\nFor\n\\(s>0\\),\nchain and product differentiation give
\n\\[\n \\eta^{(r)}(s)=E(-1/s)P_r(1/s),\\qquad\n P_0(u)=1,\\quad\n P_{r+1}(u)=u^2[P_r(u)-P_r'(u)].\n\\tag{OC39}\n\\]
\nThis exact recursion retains every polynomial coefficient, sign and\npower. If\n\\(u>0\\)\nand\n\\(m\\ge0\\)\nis an integer, the positive series gives\n\\(E(u)\\ge u^{m+1}/(m+1)!\\),\nwhence
\n\\[\n 0\\le u^mE(-u)\n \\le\\frac{(m+1)!}{u}\\longrightarrow0\n \\quad(u\\longrightarrow+\\infty).\n\\tag{OC40}\n\\]
\nEvery term of\n\\(P_r(u)\\),\nand of\n\\(uP_r(u)\\),\ntherefore tends to zero after multiplication by\n\\(E(-u)\\).\nIt follows first that\n\\(\\eta\\)\nis continuous at zero. Inductively, extend the positive-side\n\\(r\\)-th\nderivative by zero for\n\\(s\\le0\\).\nIt is continuous at zero, and its derivative there is the limit of\n\\(E(-1/s)P_r(1/s)/s\\),\nwhich is zero by the just-proved estimate. Its derivative away from zero\nis precisely the next recursion. This proves\n\\(\\eta\\in C^\\infty(\\mathbb R)\\)\nand every derivative zero at the original endpoint.
\nThe denominator in\n\\(\\chi\\)\nis strictly positive for every real\n\\(s\\).\nOtherwise both terms would vanish, requiring simultaneously\n\\(s\\ge3/2\\)\nand\n\\(s\\le1\\).\nThe full reciprocal and product rules show that\n\\(\\chi\\)\nis smooth, with\n\\(0\\le\\chi\\le1\\),\n\\(\\chi=1\\)\nfor\n\\(s\\le1\\),\nand\n\\(\\chi=0\\)\nfor\n\\(s\\ge3/2\\).\nThe function\n\\(\\chi\\)\nitself is not asserted to have compact support on the whole real line:\nits negative half-line is one. On\n\\([0,\\infty)\\)\nits support is exactly\n\\([0,3/2]\\).
\nThe actual Euclidean cutoff is\n\\(\\theta(x)=\\chi(\\sum_{j=1}^d x_j^2)\\).\nThe inner function is a polynomial in every original coordinate. The\nproved full chain rules therefore make\n\\(\\theta\\)\nsmooth; it equals one when\n\\(\\sum_jx_j^2\\le1\\)\nand vanishes when\n\\(\\sum_jx_j^2\\ge3/2\\).\nIts support is the full closed ball\n\\(\\sum_jx_j^2\\le3/2\\),\ncompact by the proved original topology. Every derivative is continuous\non that support and zero outside it; compact extrema give its actual\nfinite derivative bounds. In dimension zero the entire space is a point,\nthe value is one and its support is compact; no nonexistent coordinate\nderivative is assigned.
\nFor any specified original center\n\\(x_0\\)\nand radii\n\\(0\\le r<R\\),\nthe map\n\\[\n x\\longmapsto\n \\chi\\left(1+\\frac{\\sum_{j=1}^d(x_j-x_{0j})^2-r^2}{2(R^2-r^2)}\\right)\n\\tag{OC41}\n\\] retains every translation, square and\ndenominator. It equals one on the closed\nradius-\\(r\\)\nball. Its nonzero set is exactly the open\nradius-\\(R\\)\nball, because the full cutoff is positive precisely when its scalar\ninput is below\n\\(3/2\\).\nIts support is therefore the full closed\nradius-\\(R\\)\nball, including the outer sphere where the value itself is zero. In\ndimension zero the domain is its single point and these sets have that\nsame one-point meaning. This is a proved coordinate construction, not a\nchange to the original cutoff in (OC38). For an open domain and any one\nof its points, choose an outer ball whose closure lies inside that\ndomain; the construction gives the required local compact cutoff.
\nFor a locally finite family of supports, every point has a\nneighborhood meeting only finitely many of them. All other functions and\nall their derivatives vanish on that neighborhood. Its full sum and\nevery derivative are therefore the corresponding finite sums there. The\nsame argument applies to a compact set by its finite neighborhood\nsubcover. No interchange of an uncontrolled infinite derivative sum is\nasserted.
", "availability": "statement_and_proof_present", "related_results": [ { "id": "smooth-partition-and-cutoffs", "label": "The general partition and cutoff theorems" } ] }, { "id": "complete-metric-baire", "title": "Baire category for complete metric spaces", "lesson_id": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces", "course_id": "foundations-of-von-neumann-algebras", "lesson_label": "Hahn–Banach, Baire and the basic theorems on Banach spaces", "reader": "docs/courses/foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html#OA-FND-HB-03", "reader_label": "Statement and argument", "source": "docs/courses/foundations-of-von-neumann-algebras/src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "statement_html": "Theorem 3.1 (Baire). Let \\((X,d)\\) be a nonempty complete metric space, and \\((U_n)_{n\\ge1}\\) a sequence of dense open subsets. Then \\(\\bigcap_nU_n\\) is dense. Equivalently, \\(X\\) is not a countable union of closed sets with empty interior.
", "availability": "statement_and_proof_present" }, { "id": "hilbert-cauchy-schwarz", "title": "Cauchy–Schwarz for positive semidefinite forms", "lesson_id": "hilbert-spaces-and-compact-operators", "course_id": "foundations-of-von-neumann-algebras", "lesson_label": "Hilbert spaces and compact operators", "reader": "docs/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#OA-FND-HS-01", "reader_label": "Statement and argument", "source": "docs/courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md", "source_sha256": "833a2d60bb99e63875ce10bf363953c0bb265f92996ecc8648fc8c6a105cb1fe", "reader_sha256": "27be849d0131cecb55b41326a21d5aea490064536645e9e92ef4b13756d8a827", "statement_html": "A sesquilinear form on a complex vector space \\(V\\) is a map \\(B:V\\times V\\to\\mathbb C\\), linear in the first variable and conjugate-linear in the second. It is Hermitian if \\(B(\\eta,\\xi)=\\overline{B(\\xi,\\eta)}\\), and positive semidefinite if \\(B(\\xi,\\xi)\\ge0\\) for all \\(\\xi\\).
\nProposition 1.1.
\nLemma (Completion). A normed space \\(E\\) embeds linearly and isometrically as a dense subspace of a Banach space \\(\\widehat E\\). If its norm comes from an inner product, that product extends to \\(\\widehat E\\), which is then a Hilbert space. Every bounded linear map \\(T:E\\to F\\), with \\(F\\) a Banach space, has a unique bounded linear extension to \\(\\widehat E\\), of the same norm. In particular the Hilbert construction applies to a positive semidefinite form after taking its quotient by the null subspace.
", "availability": "statement_and_proof_present" }, { "id": "hilbert-projection-and-riesz", "title": "Hilbert projection and Riesz–Fréchet representation", "lesson_id": "hilbert-spaces-and-compact-operators", "course_id": "foundations-of-von-neumann-algebras", "lesson_label": "Hilbert spaces and compact operators", "reader": "docs/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#OA-FND-HS-02", "reader_label": "Statement and argument", "source": "docs/courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md", "source_sha256": "833a2d60bb99e63875ce10bf363953c0bb265f92996ecc8648fc8c6a105cb1fe", "reader_sha256": "27be849d0131cecb55b41326a21d5aea490064536645e9e92ef4b13756d8a827", "statement_html": "Theorem 2.1. Let \\(C\\) be a nonempty closed convex subset of a real or complex Hilbert space \\(H\\), and \\(\\xi\\in H\\). There is exactly one \\(c_0\\in C\\) with \\(\\|\\xi-c_0\\|=\\operatorname{dist}(\\xi,C)\\).
\nTheorem 2.2 (projection theorem). Let \\(L\\) be a closed subspace of a real or complex Hilbert space \\(H\\). Then:
\nTheorem 2.3 (Riesz–Fréchet). Every bounded linear functional \\(f\\) on \\(H\\) is \\(f(\\xi)=\\langle\\xi,\\eta\\rangle\\) for exactly one \\(\\eta\\in H\\), and \\(\\|f\\|=\\|\\eta\\|\\).
", "availability": "statement_and_proof_present" }, { "id": "HB-finite-dimensional-Hausdorff-vector-spaces", "title": "Finite-dimensional Hausdorff vector spaces", "lesson_id": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces", "course_id": "foundations-of-von-neumann-algebras", "lesson_label": "Hahn–Banach, Baire and the basic theorems on Banach spaces", "reader": "docs/courses/foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html#OA-FND-HB-07", "reader_label": "Statement and argument", "source": "docs/courses/foundations-of-von-neumann-algebras/src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "statement_html": "Theorem 7.1. Let \\(X\\) be a Hausdorff topological vector space of finite dimension \\(n\\). Then every linear bijection \\(f:\\mathbb K^n\\to X\\) is a homeomorphism, where \\(\\mathbb K^n\\) has its Euclidean topology. Consequently:
\nContaining lemma: full statement. The application below uses the complete-metric argument in part (3). The Polish-space conclusion also assumes separability.
Lemma 1.3.
\n