--- 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" canonical: https://lightmysky.com/learn/mathematics/lp-spaces-and-the-inequalities-they-rest-on-mt_Qd8jU26qgE source: https://lightmysky.com/learn/mathematics/lp-spaces-and-the-inequalities-they-rest-on-mt_Qd8jU26qgE.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`. # 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 - Explain why, for real Lp on [0, 1] with Lebesgue measure and finite p >= 1, only p = 2 gives an inner-product norm ## Lesson: How big is a function We use real functions on intervals of positive length with Lebesgue measure and finite p >= 1. The Lp norm of f is the p-th root of the integral of |f|^p. A constant c on a set of measure m has norm |c| times m^(1/p), so an indicator's L1 norm is its set's measure. Minkowski says the norm of f + g is at most the norm of f plus the norm of g. This needs proof because a p-th root does not split over a sum. **Example.** Holder's inequality says the integral of |fg| is at most the p-norm of f times the q-norm of g, when 1/p + 1/q = 1. Take p = q = 2 on [0, 1], with f = 2 and g the indicator of [0, 1/4]. The norm of f is 2 and the norm of g is the square root of 1/4, which is 1/2, so the bound is 1. The true integral of fg is 2 times 1/4, which is 1/2. *(drawing: The Holder bound 1 sits above the true integral 1/2.)* Two functions differing only on a null set have a difference with norm 0. An indicator of a point or countable set is nonzero there but has integral 0. Definiteness means that only the zero vector has norm 0, so Lp identifies functions equal almost everywhere. Homogeneity means that multiplying any function by c multiplies its norm by |c|. For real functions on an interval with Lebesgue measure, L2 has inner product integral of fg; zero inner product means orthogonal. Complete means every Cauchy sequence converges in the space. L2 is complete, so it is a Hilbert space. On positive-length intervals, the Lp norm for finite p >= 1 comes from an inner product only at p = 2, as the parallelogram test shows. For conjugates, solve 1/q = 1 minus 1/p: the conjugate of 4 is 4/3. An inner-product norm obeys the parallelogram law: ||f+g||^2 + ||f-g||^2 = 2||f||^2 + 2||g||^2. For real Lp on [0, 1] with Lebesgue measure and finite p >= 1, use disjoint indicators of equal length m > 0. Each norm is m^(1/p); sum and difference have norm (2m)^(1/p). The law forces 2(2m)^(2/p) = 4m^(2/p), hence p = 2. At p = 2, integral of fg gives the norm's inner product. Zero inner product means orthogonal. **Recap.** With Lebesgue measure on intervals, Holder and Minkowski govern Lp norms. Equal almost everywhere means one element; for finite p >= 1, only p = 2 gives an inner-product norm. *Original lesson, questions and flashcards authored for this dataset.* ## 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) ## Optional connections - [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)