---
title: Hilbert Spaces
description: Hilbert spaces extend the methods of vector algebra and calculus from two-dimensional and three-dimensional Euclidean space to infinite-dimensional spaces. Learners will understand inner products, orthogonality, and spectral decomposition applied to quantum mechanics and signal processing.
category: mathematics
subcategory: functional-analysis
difficulty: beginner, intermediate, advanced
url: /subject/hilbert-spaces
---

# Hilbert Spaces

Hilbert spaces extend the methods of vector algebra and calculus from two-dimensional and three-dimensional Euclidean space to infinite-dimensional spaces. Learners will understand inner products, orthogonality, and spectral decomposition applied to quantum mechanics and signal processing.

## Available Resources

1 Books • 2 Websites

## Websites

### 1. nLab: Hilbert Space

Advanced reference with category theory perspective, linked in r/math discussions.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/Hilbert+space

**Tags:** hilbert-spaces, functional-analysis, category-theory, inner-product-spaces

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Books

### 1. An Introduction to Hilbert Space

**Author:** Nicholas Young

Nicholas Young's undergraduate text develops inner product spaces, orthogonal expansions, and Fourier series, then turns to linear operators, Sturm-Liouville systems, and Hardy spaces. Readers finish able to apply Hilbert space methods to differential equations and approximation problems.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0521337178?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, functional-analysis

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