CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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On an infinite-dimensional normed space, the identity operator is not compact

Statement

Let $X$ be a normed space that admits no ordered basis of finite length, and let $I_{X} : X \to X$ be the identity map. Then $I_{X}$ is bounded, but it does not carry the closed unit ball of $X$ to a compact subset of $X$ . In that standard sense, the identity operator is not compact.

Facts & Assumptions

Given: A normed space $X$ with no ordered basis of finite length, its closed unit ball ${\overline{B}}_{X}$ , and the identity map $I_{X}$ .

[L1]

A bounded linear operator is a linear map satisfying one global norm bound (A bounded linear operator between normed spaces).

[L2]

In this setting the closed unit ball is not compact (In an infinite-dimensional normed space the closed unit ball is not compact).

Proof

technique · direct

1.1

The identity map is linear and satisfies $∥ I_{X} x ∥ = ∥ x ∥$ for every $x \in X$ , so [L1] makes it a bounded linear operator.

L1algebra

2.1

One has $I_{X} [{\overline{B}}_{X}] = {\overline{B}}_{X}$ . By [L2], that set is not compact. Therefore the identity operator does not send the closed unit ball to a compact subset of $X$ .

L2step 1.1∎

Remarks

This item uses only the unit-ball criterion. It does not depend on a separate compact-operator definition item.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)