TheoremStatement: AI-adaptedProof: AI-generated

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A bounded linear functional on an arbitrary subspace extends with the same norm, without assuming the subspace is closed

Statement

Assume the Axiom of Choice. Let $X$ be a normed space over $R$ or $C$ , let $M \subseteq X$ be a linear subspace, and let $f_{0} : M \to R$ or $f_{0} : M \to C$ be a bounded linear functional over the ambient scalar field. Then there exists a bounded linear extension $F$ of $f_{0}$ to all of $X$ such that $∥ F ∥ = ∥ f_{0} ∥$ .

No closedness hypothesis on $M$ is needed.

Facts & Assumptions

Given: The Axiom of Choice, A normed space $X$ over $R$ or $C$ , a linear subspace $M \subseteq X$ , and a bounded linear functional $f_{0}$ on $M$ .

[A1]

AC is used through [L1] or [L2] in steps 1.1 and 1.2.

[L1]

In the real case, a bounded linear functional on a subspace extends with the same norm (A bounded real linear functional on a subspace of a real normed space extends with the same norm).

[L2]

In the complex case, a bounded linear functional on a subspace extends with the same norm (A bounded complex linear functional on a subspace of a complex normed space extends with the same norm).

[L3]

The page's normed-space language is read over $R$ by A norm on a real vector space, the induced metric, and the dictionary with the metric axioms and over $C$ by Complex norm and complex normed space.

Proof

technique · direct

1.1L1L3given

If the scalar field is $R$ , then [L1] applies exactly as stated to the given subspace $M$ and produces a norm-preserving extension of $f_{0}$ to $X$ .

1.2L2L3given

If the scalar field is $C$ , then [L2] applies exactly as stated to the given subspace $M$ and produces a norm-preserving extension of $f_{0}$ to $X$ .

2.1step 1.1step 1.2L1L2∎

Neither step 1.1 nor step 1.2 uses or requires that $M$ be closed; each cited theorem assumes only that $M$ is a linear subspace. Therefore every bounded linear functional on an arbitrary subspace extends with the same norm.

Depends on

Used by

Finite-dimensional subspaces are complemented Corollary
Membership in the transpose range by an operator estimate Lemma
Geometric Hahn--Banach theorem for subspaces Theorem
The dual of a closed subspace is a dual quotient Theorem

Dependency tree · two levels

28 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

Daniel Daners, Introduction to Functional Analysis, Theorem 26.7 (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis, Corollary 4.15 and Theorem 4.14 (standard reference, not scraped)