LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Fenchel–Young equality for a Banach regularizer

Statement

Let $X$ be a real Banach space, let $R : X \to (- \infty, + \infty]$ be proper and convex, and define its Fenchel conjugate $R^{*} : X^{*} \to (- \infty, + \infty]$ by $R^{*} (ξ) : = \sup_{z \in X} (⟨ ξ, z ⟩_{X^{*}, X} - R (z))$ . For every $x \in dom R$ and $ξ \in X^{*}$ , the Fenchel–Young inequality and its equality characterization are

$R (x) + R^{*} (ξ) \geq ⟨ ξ, x ⟩_{X^{*}, X}, R (x) + R^{*} (ξ) = ⟨ ξ, x ⟩_{X^{*}, X} ⟺ ξ \in \partial R (x) .$

Statement references

Jonathan M. Borwein and Jon D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Proposition 4.4.1(a)

Facts & Assumptions

Given: $X$ , $R$ , $R^{*}$ , $x$ , and $ξ$ satisfy the hypotheses and definitions in the Statement.

[F1]

Banach-space convex subgradients and dual pairings: $ξ \in \partial R (x)$ exactly when $R (z) \geq R (x) + ⟨ ξ, z - x ⟩_{X^{*}, X}$ for every $z \in X$ .

Proof

1.1givenalgebra

The supremum defining $R^{*} (ξ)$ includes the term at $z = x$ , so $R^{*} (ξ) \geq ⟨ ξ, x ⟩_{X^{*}, X} - R (x)$ . Adding $R (x)$ gives the Fenchel–Young inequality.

2.1givenF1step 1.1

If $ξ \in \partial R (x)$ , [F1] gives $⟨ ξ, z ⟩_{X^{*}, X} - R (z) \leq ⟨ ξ, x ⟩_{X^{*}, X} - R (x)$ for all $z$ ; taking the supremum and then using the term $z = x$ proves equality.

3.1givenF1algebra∎

Conversely, if equality holds, every term in the supremum satisfies $⟨ ξ, z ⟩_{X^{*}, X} - R (z) \leq R^{*} (ξ) = ⟨ ξ, x ⟩_{X^{*}, X} - R (x)$ . Rearrangement gives [F1], hence $ξ \in \partial R (x)$ .

Depends on

Banach-space convex subgradients and dual pairings

Used by

Bregman distances for convex regularizers Definition
Dual attainment and convergence of TV source certificates Lemma

Dependency tree · two levels

5 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

Jonathan M. Borwein and Jon D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Proposition 4.4.1(a) (standard reference, not scraped)