uniform boundedness principle

一致有界原理

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Translation evidence: Literature usage

Meaning in context

For a family of bounded linear maps from a Banach space X to a normed space Y, if sup_T norm(Tx) is finite for every x in X, then sup_T norm(T) is finite.

从 Banach 空间到赋范空间的一族有界线性算子，若在每个固定向量上像的范数一致有界，则整族算子范数一致有界。

Example

On $\ell^2$, the functionals $T_nx=x_n$ are pointwise bounded by $\|x\|_2$ and all have operator norm $1$.

Usage and distinctions

Pointwise bounds may depend on x; the conclusion gives one operator-norm bound for the entire family, using completeness of X.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§6.7, Definitions 6.7.1–6.7.2 and principle stated after Exercise 6.7.3 · Checked 2026-10-14

Translation sources

1. 电子科技大学 — 泛函分析教学大纲第2章，了解2：一致有界原理 · Checked 2026-10-14

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