Hilbert space

Hilbert空间

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

Translation evidence: Literature usage

Meaning in context

A Hilbert space is a real or complex inner-product space complete for the norm ⟨x,x⟩\sqrt{\langle x,x\rangle} induced by that inner product.

Hilbert 空间是由内积诱导的范数下完备的实或复内积空间；复内积须指定共轭线性所在的变量。

Example

ℓ2\ell^2 with ⟨x,y⟩=∑xnyn‾\langle x,y\rangle=\sum x_n\overline{y_n} is Hilbert; ∥(1,2,0,…)∥=5\|(1,2,0,\ldots)\|=\sqrt5.

Usage and distinctions

A Banach space needs only a complete norm; a Hilbert norm also obeys the parallelogram identity.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§4.1, Definition 4.1.1 and Example 4.1.3 · Checked 2026-10-14

Translation sources

  1. 电子科技大学 — 泛函分析教学大纲第一章，1.3 Hilbert空间；掌握6：正交分解 · Checked 2026-10-14
Edit source / propose a PRReport a correction

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.