---
title: "Bounded Linear Operators and the Operator Norm"
description: "For a linear map between normed spaces, continuity and boundedness turn out to be the same condition. The operator norm records the worst stretching the map can perform on a unit vector."
canonical: https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx
source: https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx.md
retrieved: 2026-11-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Bounded Linear Operators and the Operator Norm

For a linear map between normed spaces, continuity and boundedness turn out to be the same condition. The operator norm records the worst stretching the map can perform on a unit vector.

Subject: Mathematics · Area: Calculus & Analysis · Ages 23 to 24
Page: https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx

## Ready when they can

- Prove that a linear map is continuous exactly when it is bounded
- Compute the operator norm of a multiplication or integral operator
- Give an unbounded linear map and explain why differentiation is the standard example
- State submultiplicativity for bounded operators: ||ST|| <= ||S|| ||T||.

## Lesson: How much can a linear map stretch

The operator norm of a linear map T is the least upper bound of ||Tx|| over all unit vectors x. T is bounded when this number is finite, and then ||Tx|| is at most ||T|| times ||x||. Bounded implies continuous, since ||Tx minus Ty|| is at most ||T|| times ||x minus y||. Conversely, if T is continuous at 0, some ball of radius r maps into the unit ball, and scaling gives ||T|| at most 1/r.

In a finite-dimensional space, fix a basis and T becomes a matrix whose columns are the images of the basis vectors. Then ||Tx|| is at most the sum of the column lengths times the largest absolute coordinate of x, and norm equivalence bounds that magnitude by a constant times ||x||. So every linear map there is bounded. Differentiation is different: on polynomials with the sup norm on [0, 1], x to the n has norm 1 but its derivative has norm n.

*(drawing: Each x to the n has sup norm 1, but its derivative has sup norm n, so no single bound works.)*

**Example.** Multiplication by g sends f to gf. On L2 of [0, 1], take g = 3x. Since |gf| <= 3|f|, the operator norm is at most 3. Functions concentrated near x = 1 are stretched by almost 3, so it is exactly 3. Rotations preserve length. Nonzero orthogonal projections never stretch and fix nonzero vectors in their range, so both have norm 1. Cauchy-Schwarz bounds a dot product by the product of lengths and helps estimate operator norms.

Chaining bounded maps multiplies their bounds at worst. For a unit vector x, ||STx|| is at most ||S|| times ||Tx||, which is at most ||S|| times ||T||. So ||ST|| is at most ||S|| ||T||, which is submultiplicativity.

**Recap.** Take the supremum over unit inputs. For linear maps, boundedness and continuity are equivalent. Test differentiation using x to the n.

*Original lesson, questions and flashcards authored for this dataset.*

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Normed Spaces, Completeness and Banach Spaces](https://lightmysky.com/learn/mathematics/normed-spaces-completeness-and-banach-spaces-mt_y_jueKnovX)

## Optional connections

- [Linear Maps and Their Matrices](https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto)

## Opens up

- [Baire Category and the Uniform Boundedness Principle](https://lightmysky.com/learn/mathematics/baire-category-and-the-uniform-boundedness-principle-mt_11Rd73trAX)
- [Unitary Time Evolution and the Heisenberg Picture](https://lightmysky.com/learn/science/unitary-time-evolution-and-the-heisenberg-picture-mt_3JsTc9VYj8)
- [The Hahn-Banach Theorem and the Dual Space](https://lightmysky.com/learn/mathematics/the-hahn-banach-theorem-and-the-dual-space-mt_pJuHFxkvjS)
- [Convolution, Plancherel and Transforming a Derivative](https://lightmysky.com/learn/mathematics/convolution-plancherel-and-transforming-a-derivative-mt_zSwMJrQIMt)