TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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A Schauder basis implies the bounded approximation property

Statement

Assume DC. If a Banach space $X$ has a Schauder basis with basis constant $K$ , then $X$ has $K$ -BAP, and hence AP.

Facts & Assumptions

[A1]

DC holds (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[L1]

Under DC, the partial-sum projections are bounded and satisfy $\sup_{N} ∥ P_{N} ∥ = K < \infty$ (Coordinate functionals of a Schauder basis are bounded).

[L2]

By the defining expansion of a Schauder basis, $P_{N} x \to x$ for every $x \in X$ (Schauder basis and coordinate functionals).

[L3]

Uniformly bounded pointwise-convergent bounded operators converge uniformly on compact sets (Uniformly bounded pointwise-convergent operators converge uniformly on compact sets).

[L4]

$K$ -BAP is compact-uniform approximation of the identity by finite-rank maps of norm at most $K$ (Approximation property and bounded approximation property).

Proof

technique · direct

Given: The objects and hypotheses in the Statement.

1.1A1L1L2

Each $P_{N}$ has range in $span {e_{1}, \dots, e_{N}}$ and hence [given, L1, A1, L2] has finite rank. By [L1], using [A1] exactly through the coordinate-boundedness theorem, $∥ P_{N} ∥ \leq K$ ; by [L2], $P_{N} x \to x$ for every $x \in X$ .

2.1L3L4step 1.1∎

Apply [L3] to $(P_{N})$ and the identity. On each compact $C$ , [given, L3, L4, step 1.1] $\sup_{x \in C} ∥ P_{N} x - x ∥ \to 0$ . Together with step 1.1, [L4] says precisely that $X$ has $K$ -BAP. Since BAP implies AP by [L4], the consequence follows.

Depends on

Used by

A separable reflexive Banach space without the approximation property Theorem

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Thomas Schlumprecht, Course Notes in Functional Analysis, Math 655 (standard reference, not scraped)