continuous dual space

对偶空间

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Translation evidence: Literature usage

Meaning in context

The continuous dual X* of a normed space X is the vector space of bounded linear scalar-valued functionals on X, equipped with the operator norm.

赋范空间 X 的对偶空间，由 X 上所有有界线性标量泛函组成，并配以算子范数。

Example

For ℓ2\ell^2, λ(x)=x3\lambda(x)=x_3 belongs to the continuous dual and has norm 11.

Usage and distinctions

The algebraic dual contains all linear functionals, including discontinuous ones in infinite-dimensional normed spaces.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§3.2, Example 3.2.9 · Checked 2026-10-14

Translation sources

  1. 电子科技大学 — 泛函分析教学大纲第2章，2.2 对偶与Hahn-Banach定理；了解1：对偶空间 · Checked 2026-10-14
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