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CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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A separately continuous bilinear map on Banach spaces is jointly continuous

Statement

Assume DC. If X,Y are Banach, Z normed, and b:X×Y→Z is bilinear and separately continuous, then b is jointly continuous.

Facts & Assumptions

Given: DC, Banach X,Y, normed Z, and separately continuous bilinear b.

Proof

technique · direct
1.1

For each y in the unit ball of Y, x↦b(x,y) is bounded; for fixed x, separate continuity makes their values at x bounded on that unit ball.

given
2.1

Uniform boundedness Uniform boundedness principle gives C with ∥b(x,y)∥≤C∥x∥∥y∥.

step 1.1
3.1

Thus b is bounded in the sense of A bounded bilinear map between normed spaces, and the boundedness/joint-continuity equivalence For a bilinear map, boundedness is equivalent to joint continuity yields joint continuity.

step 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources