---
tag: 4DJ4V5VBSH
kind: concept
name: Normed algebra
msc: [46H05]
sources:
  - url: https://en.wikipedia.org/wiki/Normed_algebra
same_as:
  wikidata: Q1999917
  wikipedia: Normed algebra
---

## Summary

A normed algebra is an algebra over the real or complex numbers with a norm that is compatible with the product.
The norm of a product is at most the product of the norms.
This inequality makes multiplication continuous and lets power series of elements converge.
A complete normed algebra is a Banach algebra.
Algebras of bounded operators and of bounded continuous functions are the main examples.

## Explanations

### L1 human

A normed algebra is a space where one can add, scale and multiply elements, and also measure their size.
The rule $\norm{xy} \le \norm{x} \norm{y}$ says that multiplying cannot blow sizes up more than the sizes themselves allow.
For square matrices with the operator norm, this rule says that applying two maps in a row stretches vectors by at most the product of the two stretch factors.