normed vector space

赋范线性空间

Analysis, Measure & PDE · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A normed vector space is a real or complex vector space equipped with a specified norm, inducing the distance d(x,y)=norm(x-y).

赋范线性空间是配有指定范数的实或复线性空间，其距离由两向量之差的范数给出。

Example

Finite-support sequences with ∥x∥2=(∑∣xn∣2)1/2\|x\|_2=(\sum |x_n|^2)^{1/2} form a normed space but are incomplete.

Usage and distinctions

A vector space alone specifies algebraic operations; a Banach space additionally requires completeness for its norm.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§1.2, Definition 1.2.13 and following normed-space paragraph · Checked 2026-10-14

Translation sources

  1. 电子科技大学 — 泛函分析教学大纲第一章赋范线性空间；了解2 · Checked 2026-10-14
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