Functional analysis

Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows. One may also speak of quadratic integrability over bounded intervals such as for . An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of square integrable functions (with respect to Lebesgue measure) forms the Lp space with . Among the Lp spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the Lp spaces are complete under their respective p-norms. Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere. (Wikipedia).

Video thumbnail

(New Version Available) Inverse Functions

New Version: https://youtu.be/q6y0ToEhT1E Define an inverse function. Determine if a function as an inverse function. Determine inverse functions. http://mathispower4u.wordpress.com/

From playlist Exponential and Logarithmic Expressions and Equations

Video thumbnail

Math 023 Fall 2022 113022 Inverse Trigonometric Functions

Recall: what functions are invertible? Functions that pass the "horizontal line test" - that is, one-to-one functions. Is the sine function invertible? No. But if we restrict it to, say [-pi/2, pi/2], then it is invertible. So we define the inverse sine function to be the inverse of t

From playlist Course 1: Precalculus (Fall 2022)

Video thumbnail

Math 023 Precalculus Day 16: Linear and Quadratic Functions

Linear functions: definition, examples (linear depreciation, cost function, marginal cost, comment about rate of change, velocity). Quadratic functions: using completing the square to simplify the expression; finding the vertex, axis of symmetry, etc.

From playlist Course 1: Precalculus

Video thumbnail

Calculus 1.5 Inverse Functions and Logarithms

My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart

From playlist Calculus

Video thumbnail

Math 030 Calculus I 031315: Inverse Functions and Differentiation

Inverse functions. Examples of determining the inverse. Relation between the graphs of a function and its inverse. One-to-one functions. Restricting the domain of a function so that it is invertible. Differentiability of inverse functions; relation between derivatives of function and

From playlist Course 2: Calculus I

Video thumbnail

Graphing Squared Functions (2 of 2: Cubic + semi-circle examples)

More resources available at www.misterwootube.com

From playlist Further Work with Functions

Video thumbnail

Abstract Algebra | Injective Functions

We give the definition of an injective function, an outline of proving that a given function is injective, and a few examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Abstract Algebra

Video thumbnail

Definition of a Surjective Function and a Function that is NOT Surjective

We define what it means for a function to be surjective and explain the intuition behind the definition. We then do an example where we show a function is not surjective. Surjective functions are also called onto functions. Useful Math Supplies https://amzn.to/3Y5TGcv My Recording Gear ht

From playlist Injective, Surjective, and Bijective Functions

Video thumbnail

Completeness and Orthogonality

A discussion of the properties of Completeness and Orthogonality of special functions, such as Legendre Polynomials and Bessel functions.

From playlist Mathematical Physics II Uploads

Video thumbnail

Calculus 15.2 Double Integrals over General Regions

My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart

From playlist Calculus

Video thumbnail

Calculus 15.6 Triple Integrals

My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart

From playlist Calculus

Video thumbnail

What is General Relativity? Lesson 35: Math Break: Integration/Radicals

What is General Relativity? Lesson 34: Math Break - Integration of Rational Functions involving the square root of a quadratic, This is another mathematics review to prepare for the introduction of elliptic integrals. The paper by Kostic I occasionally refer to during these lectures can

From playlist What is General Relativity?

Video thumbnail

Lec 29 | MIT 18.085 Computational Science and Engineering I, Fall 2008

Lecture 29: Fourier series (part 2) License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 18.085 Computational Science & Engineering I, Fall 2008

Video thumbnail

Calculus 15.4 Applications of Double Integrals

My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart

From playlist Calculus

Video thumbnail

Mod-01 Lec-11 Integration II | Biomathematics by IIT Bombay

Biomathematics by Dr. Ranjith Padinhateeri, Department of Biotechnology, IIT Bombay.For more details on NPTEL visit http://nptel.iitm.ac.in

From playlist IIT Bombay: Biomathematics | CosmoLearning.org Bioengineering

Video thumbnail

Calculus 2 - Full College Course

Learn Calculus 2 in this full college course. This course was created by Dr. Linda Green, a lecturer at the University of North Carolina at Chapel Hill. Check out her YouTube channel: https://www.youtube.com/channel/UCkyLJh6hQS1TlhUZxOMjTFw ⭐️ Prerequisites ⭐️ 🎥 Algebra: https://www.yout

From playlist Mathematics

Video thumbnail

Worldwide Calculus: Integration in R²

Lecture on 'Integration in R²' from 'Worldwide Multivariable Calculus'. For more lecture videos and $10 digital textbooks, visit www.centerofmath.org.

From playlist Multivariable Integrals

Video thumbnail

Area Between Curves #2 (OpenStax Calculus, Vol. 2, Section 2.1)

This video contains solutions to sample problems from OpenStax Calculus, Volume 2, Section 2.1: Area Between Curves. This is the second of two videos, focusing on more complicated examples. OpenStax Calculus Vol. 2: https://openstax.org/details/books/calculus-volume-2

From playlist Calculus II

Video thumbnail

Given a quadratic function, find the inverse and determine if a function or not

👉 Learn how to find the inverse of a quadratic function. A quadratic function is a function whose highest exponent in the variable(s) of the function is 2. The inverse of a function is a function that reverses the "effect" of the original function. One important property of the inverse of

From playlist Find the Inverse of a Function

Video thumbnail

What is General Relativity? Lesson 34: Math Break - Integration of Rational Functions

What is General Relativity? Lesson 34: Math Break - Integration of Rational Functions This is another mathematics review to prepare for the introduction of elliptic integrals. The paper by Kostic I occasionally refer to during these lectures can be found here: https://arxiv.org/abs/1201

From playlist What is General Relativity?

Related pages

Complex conjugate | Integral | Hilbert space | Equivalence class | Absolute value | Banach space | Cauchy space | Complex number | Mathematics | Lp space | Measurable function | Inner product space | Real number | Almost everywhere | Complete metric space | Cauchy sequence