Uniform boundedness principle

If a family of bounded linear operators from a Banach space to a normed space is pointwise bounded, then their operator norms are uniformly bounded.

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- Uniform bound from pointwise absolute summability of dual evaluations Uniform boundedness principle

Uniform bound from pointwise absolute summability of dual evaluations

Let

(x_{n})

lie in a normed space

X

and suppose

\sum_{n} ∣ f (x_{n}) ∣ < \infty

for every

f \in X^{*}

. Then some

C

satisfies

\sum_{n} ∣ f (x_{n}) ∣ \leq C ∥ f ∥ (f \in X^{*}) .

Apply the uniform boundedness principle on the Banach space

X^{*}

to the finite signed sums

f \mapsto \sum_{n} a_{n} f (x_{n})

with

∣ a_{n} ∣ \leq 1

.

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codex/banach-steinhaus-theorem

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The Uniform Boundedness Principle, also known as the Banach-Steinhaus theorem, is a fundamental result in functional analysis. It provides conditions under which a family of bounded linear operators is uniformly bounded.

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