---
title: "Lp Spaces and the Inequalities They Rest On"
description: "Functions with integrable p-th power form a normed space once functions that agree almost everywhere are treated as equal. Holder's and Minkowski's inequalities are what make the norm behave like a le"
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source: https://lightmysky.com/learn/mathematics/lp-spaces-and-the-inequalities-they-rest-on-mt_Qd8jU26qgE.md
retrieved: 2026-09-12
---

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# Lp Spaces and the Inequalities They Rest On

Functions with integrable p-th power form a normed space once functions that agree almost everywhere are treated as equal. Holder's and Minkowski's inequalities are what make the norm behave like a length.

Subject: Mathematics · Area: Calculus & Analysis · Ages 22 to 24
Page: https://lightmysky.com/learn/mathematics/lp-spaces-and-the-inequalities-they-rest-on-mt_Qd8jU26qgE

## Ready when they can

- Use Holder's inequality to bound the integral of a product
- Explain the identification of functions agreeing almost everywhere and why the norm forces it
- Say what makes the case p equal to 2 different from every other p

## Lesson: How big is a function

You measure a function by its Lp norm: integrate its p-th power and take the p-th root. Minkowski proved the triangle inequality for this measurement, which is why the space of functions with finite norm behaves like a geometry with lengths. Holder bounds the integral of a product by the product of the norms, using conjugate exponents whose reciprocals sum to 1.

**Example.** The indicator of the interval from 1 to 4 has L1 norm equal to its length, which is 3. The constant 3 on an interval of length 4 has L2 norm equal to the square root of 9 times 4, which is 6. You compute both directly from the definition of the norm.

You must treat two functions that agree except on a null set as the same element of Lp. The reason is definiteness: the indicator of a single point is nonzero somewhere yet its integral is 0, so without this step a nonzero function would carry norm 0. Identifying almost everywhere equal functions repairs the norm.

**Tip.** When p equals 2, the norm comes from an inner product, so you gain orthogonality and the parallelogram law, and the space is a Hilbert space. For every other exponent no inner product sits behind the norm. As a quick check for Holder, the conjugate of 4 is 4 over 3, since 1 over 4 plus 3 over 4 equals 1.

**Recap.** Holder and Minkowski make Lp a normed geometry, almost everywhere equal functions count as one element, and only p equal to 2 brings an inner product.

## Practice

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

## Needs first

- [Product Measure and Fubini's Theorem](https://lightmysky.com/learn/mathematics/product-measure-and-fubinis-theorem-mt_1VzzVmMjTB)
- [The Lebesgue Integral and What It Repairs](https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU)
- [Inner Products, Length and Orthogonality](https://lightmysky.com/learn/mathematics/inner-products-length-and-orthogonality-mt_X0HPRGto4W)

## Opens up

- [Weak Solutions and Test Functions](https://lightmysky.com/learn/mathematics/weak-solutions-and-test-functions-mt_AKlC3Ol70p)
- [The Fourier Transform on the Line](https://lightmysky.com/learn/mathematics/the-fourier-transform-on-the-line-mt_BMrt5FPe3U)
- [Normed Spaces, Completeness and Banach Spaces](https://lightmysky.com/learn/mathematics/normed-spaces-completeness-and-banach-spaces-mt_y_jueKnovX)