Sections 3 and 8 of the earlier programme lesson Banach estimates, quotient spaces and compact parameter arguments, AN03-P004. Programme credits and the CC0 dedication are retained in the accompanying notices. Selection by GPT-6 Astra (OpenAI), October 2026.
These proofs supply complete operator spaces, summable vector series and bounded inverses of Banach-space isomorphisms. The functional foundation, Section 6, supplies the complete-metric Baire theorem. Scalar completeness and geometric series are proved in the scalar foundation, Sections 12–13.
3. Completeness of operator spaces
First note that a linear map is continuous exactly when for some finite . The estimate implies continuity. Conversely, continuity at zero gives when , for some . For , apply this to to obtain . Define
It is finite, and scaling gives . The triangle inequality and homogeneity pass through the supremum. If , scaling shows that vanishes everywhere. Thus this is a norm, also for the zero domain, whose unit ball still contains zero. Composition satisfies .
A closed linear subspace of a Banach space is Banach: a Cauchy sequence has a limit in the ambient space, and closedness keeps the limit in the subspace. A finite product of Banach spaces is Banach under . A Cauchy sequence has a limit in every coordinate; taking the maximum of finitely many coordinate errors proves norm convergence. The sum norm is equivalent, since
The empty product is the zero vector space. A product of the coordinate norms is not a norm and is never used here.
If is Banach and merely normed, then is Banach. For an operator-norm Cauchy sequence , the values are Cauchy for every fixed . Let be their limit in . Passing to the limit in proves linearity. The operator norms of the sequence are bounded: the tail lies within one of a fixed operator, and the remaining initial segment is finite. Calling a bound , passage to the limit gives , so . Given , take with . Holding and fixed and taking yields
and taking the unit-ball supremum gives . The completeness of was unnecessary. In particular is Banach.
We shall repeatedly use the resulting series criterion: if is Banach and , then converges in , with every tail norm bounded by the corresponding scalar tail sum. Indeed the finite partial sums are Cauchy by the triangle inequality; pass to their limit in that same inequality. No integration theorem is involved.
8. Open mapping through summable corrections
Let be a bounded surjective linear map between Banach spaces. We show that a fixed ball about zero in lies in the image of the closed unit ball of . If , this holds for every positive radius, so suppose otherwise. Write for the open unit ball. Surjectivity gives
By the Baire theorem, Section 6, applied to , some closed set on the right contains , where . If , approximate both and by images of vectors in . Their differences approximate by images of vectors in . Consequently
The second inclusion follows by scaling. Scaling again gives for every .
Take . From (B8) with , choose with and . If have been chosen with residual norm less than , use (B8) at to choose with
Section 3 and completeness of give with . Continuity of and the residual bounds give . This is actual surjectivity of a ball, not merely density of its image.
For any open and , choose with , for example half a radius of an open ball contained in . The preceding conclusion gives
Therefore is open. If is bijective, its inverse is linear by uniqueness of preimages. For , apply the unit-ball conclusion to ; rescaling and uniqueness yield
The zero vector satisfies the same estimate. Thus the inverse is bounded. The two completeness assumptions have distinct uses: Baire on , and convergence of the correction series in .