--- title: Lp Spaces description: These function spaces are defined using a natural generalization of the Euclidean norm for integrable functions. Learners will understand the properties of Banach and Hilbert spaces, completeness, and how to analyze functions using these geometric frameworks. category: mathematics subcategory: measure-theory difficulty: beginner, intermediate, advanced url: /subject/lp-spaces --- # Lp Spaces These function spaces are defined using a natural generalization of the Euclidean norm for integrable functions. Learners will understand the properties of Banach and Hilbert spaces, completeness, and how to analyze functions using these geometric frameworks. ## Where to start Start with Measure Theory Notes, Chapter 7: Lp Spaces by John K. Hunter, a free standalone chapter covering the Hölder and Minkowski inequalities, completeness and duality. If you only use one resource, make it Real Analysis by Gerald B. Folland, a paid graduate text on measure, integration and functional analysis. For the inequalities in depth, move on to Analysis, Second Edition by Elliott H. Lieb and Michael Loss, also paid. ## Available Resources 2 Books • 1 Courses • 2 Websites • 1 Papers ## Courses ### 1. Real Analysis (MIT 18.100A) **Author:** Casey Rodriguez Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ **Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral ## Books ### 1. Analysis, Second Edition **Author:** Elliott H. Lieb, Michael Loss AMS graduate text whose Chapter 2 is devoted to Lp-spaces and whose Chapters 3 and 4 develop the inequalities that make them useful, including rearrangement, Hölder, Young and Hardy-Littlewood-Sobolev, with proofs oriented toward PDE and mathematical physics. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0821827839?tag=edmonddante07-20 **Tags:** lp-spaces, real-analysis, inequalities, rearrangement-inequalities, sobolev-spaces, measure-theory ### 2. Real Analysis: Modern Techniques and Their Applications (2nd ed.) **Author:** Gerald B. Folland The reference every graduate course cites for the abstract Fubini-Tonelli statement. Standard graduate reference on measure, integration, and functional analysis. Chapter 2 develops the abstract integral, with section 2.5 treating product measures and the Fubini-Tonelli theorem for sigma-finite spaces, including the completion subtleties on R^n. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0471317160?tag=edmonddante07-20 **Tags:** real-analysis, measure-theory, lebesgue-integration, fubini-tonelli, functional-analysis ## Websites ### 1. 245B, Notes 3: Lp Spaces **Author:** Terence Tao Graduate lecture notes from Tao's UCLA real analysis course treating Lp spaces as the model example of normed vector spaces: quasi-norms, Minkowski and Hölder inequalities, completeness as Banach spaces, density of simple functions, and Lp duality, with embedded exercises. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://terrytao.wordpress.com/2009/01/09/245b-notes-3-lp-spaces/ **Tags:** lp-spaces, real-analysis, banach-spaces, holder-inequality, duality, lecture-notes ### 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 ## Papers ### 1. Measure Theory Notes, Chapter 7: Lp Spaces **Author:** John K. Hunter Standalone chapter from a UC Davis graduate measure theory course covering the Lp norm, Hölder and Minkowski inequalities, completeness by the Riesz-Fischer argument, density of simple and continuous functions, and the duality between Lp and its conjugate exponent. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch7.pdf **Tags:** lp-spaces, measure-theory, holder-inequality, minkowski-inequality, riesz-fischer-theorem, lecture-notes --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/lp-spaces