Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-04
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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In an infinite-dimensional normed space the closed unit ball is not compact

Statement

Let X be a normed space that admits no ordered basis of finite length. Then its closed unit ball

B‾X:={x∈X:∥x∥≤1}

is not compact.

Facts & Assumptions

Given: A normed space X with no ordered basis of finite length.

[L1]

The closed unit ball is compact if and only if the space admits an ordered basis of finite length (The closed unit ball is compact if and only if the normed space is finite-dimensional).

Proof

technique · direct
1.1

If B‾X were compact, [L1] would force X to admit an ordered basis of finite length.

L1assume-contra
2.1

That contradicts the hypothesis, so B‾X is not compact.

step 1.1discharge-contradiction∎

Depends on

Used by

Dependency tree · two levels

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Sources