Generalizations of the derivative | Banach spaces

Quasi-derivative

In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux derivative, though weaker than the Fréchet derivative. Let f : A → F be a continuous function from an open set A in a Banach space E to another Banach space F. Then the quasi-derivative of f at x0 ∈ A is a linear transformation u : E → F with the following property: for every continuous function g : [0,1] → A with g(0)=x0 such that g′(0) ∈ E exists, If such a linear map u exists, then f is said to be quasi-differentiable at x0. Continuity of u need not be assumed, but it follows instead from the definition of the quasi-derivative. If f is Fréchet differentiable at x0, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at x0. The converse is true provided E is finite-dimensional. Finally, if f is quasi-differentiable, then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative. (Wikipedia).

Video thumbnail

Calculus - What is a Derivative? (1 of 8) Overview

Visit http://ilectureonline.com for more math and science lectures! In this video I will give a general overview and the definition of “What is a derivative?”

From playlist CALCULUS 1 CH 2 WHAT IS A DERIVATIVE?

Video thumbnail

Multivariable Calculus | Definition of partial derivatives.

We give the definition of the partial derivative of a function of more than one variable. In addition, we present some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Multivariable Calculus

Video thumbnail

Calculus 3: Partial Derivative (1 of 50) What is a Partial Derivative?

Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is the difference between a derivative and partial derivative, and what is the physical meaning of a partial derivative. Next video in the series can be seen at: https://youtu.be/rfX3AYN

From playlist CALCULUS 3 CH 4 PARTIAL DERIVATIVES

Video thumbnail

Calculus 1: What is a Derivative? (1 of 9) Basic Definitions

Visit http://ilectureonline.com for more math and science lectures! In this video I will introduce the basic definitions of what is a derivative. Next video in the series can be seen at: http://youtu.be/29Px0qXE1BU

From playlist CALCULUS 1 CH 2 WHAT IS A DERIVATIVE?

Video thumbnail

Definition of derivative in terms of a limit, (def 1)

Definition of derivative, calculus 1 homework solution. #calculus Check out my 100 derivatives: https://youtu.be/AegzQ_dip8k

From playlist Sect 2.7, Definition of Derivative

Video thumbnail

Calculus - What is a Derivative? (7 of 8) A Function Describing a Change (In a Function)

Visit http://ilectureonline.com for more math and science lectures! In this video I will describe the physical meaning of a derivative.

From playlist CALCULUS 1 CH 2 WHAT IS A DERIVATIVE?

Video thumbnail

Calculus 3.03a - The Derived Function

The Derived Function, or better known as simply The Derivative. This is the beginning of the material on the definition of the derivative. The topic is approached in a fairly standard manner as the general case of the limit of a difference quotient.

From playlist Calculus Ch 3 - Derivatives

Video thumbnail

Étale cohomology 9/15/2020

Čech cohomology part II, Čech-to-derived spectral sequence, Mayer-Vietoris, étale cohomology of quasi-coherent sheaves, the Artin-Schreier exact sequence and the étale cohomology of F_p in characteristic p.

From playlist Étale cohomology and the Weil conjectures

Video thumbnail

Winter School JTP: Introduction to A-infinity structures, Bernhard Keller, Lecture 3

In this minicourse, we will present basic results on A-infinity algebras, their modules and their derived categories. We will start with two motivating problems from representation theory. Then we will briefly present the topological origin of A-infinity structures. We will then define and

From playlist Winter School on “Connections between representation Winter School on “Connections between representation theory and geometry"

Video thumbnail

C. Leininger - Teichmüller spaces and pseudo-Anosov homeomorphism (Part 1)

I will start by describing the Teichmuller space of a surface of finite type from the perspective of both hyperbolic and complex structures and the action of the mapping class group on it. Then I will describe Thurston's compactification of Teichmuller space, and state his classification

From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications

Video thumbnail

Higher Algebra 5: Slices and filtered colimits

In this video, we provide further properties of the derived category of an abelian category. Along the way we discuss slice categories and filtered colimits. This is the fifth video in our introduction to ∞-categories and Higher Algebra. Feel free to post comments and questions at our pub

From playlist Higher Algebra

Video thumbnail

C. Leininger - Teichmüller spaces and pseudo-Anosov homeomorphism (Part 2)

I will start by describing the Teichmuller space of a surface of finite type from the perspective of both hyperbolic and complex structures and the action of the mapping class group on it. Then I will describe Thurston's compactification of Teichmuller space, and state his classification t

From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications

Video thumbnail

B. Deroin - Monodromy of algebraic families of curves (Part 2)

The mini-course will focus on the properties of the monodromies of algebraic families of curves defined over the complex numbers. One of the goal will be to prove the irreducibility of those representations for locally varying families (Shiga). If time permit we will see how to apply this

From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications

Video thumbnail

What is the Derivative?

Derivatives are the main object of study in differential calculus. They describe rates of change of functions. That makes them incredibly useful in all of science, as many models can be expressed by describing the changes over time (e.g. of physical quantities). However, the abstract defin

From playlist Summer of Math Exposition Youtube Videos

Video thumbnail

Anna Vershynina: "Quasi-relative entropy: the closest separable state & reversed Pinsker inequality"

Entropy Inequalities, Quantum Information and Quantum Physics 2021 "Quasi-relative entropy: the closest separable state and the reversed Pinsker inequality" Anna Vershynina - University of Houston Abstract: It is well known that for pure states the relative entropy of entanglement is equ

From playlist Entropy Inequalities, Quantum Information and Quantum Physics 2021

Video thumbnail

Massimiliano BERTI - Quasi - periodic standing wave solutions of gravity-capillary water waves

We prove the existence of Cantor families of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x ) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. In a

From playlist Trimestre "Ondes Non Linéaires" - May Conference

Video thumbnail

How to solve quasi linear PDE

Free ebook https://bookboon.com/en/partial-differential-equations-ebook How to solve quasi linear PDE. I discuss and solve an example.

From playlist Partial differential equations

Video thumbnail

derivative of x^-2 with the definition of derivative

We use the definition of derivative to find the derivative of x^-2. For more calculus tutorials, please see my new "just calculus" channel: 👉https://www.youtube.com/justcalculus If you find my videos helpful, then consider supporting me on Patreon: 👉 https://www.patreon.com/blackpenred

From playlist Sect 2.7, Definition of Derivative

Video thumbnail

Jacob Lurie: A Riemann-Hilbert Correspondence in p-adic Geometry Part 2

At the start of the 20th century, David Hilbert asked which representations can arise by studying the monodromy of Fuchsian equations. This question was the starting point for a beautiful circle of ideas relating the topology of a complex algebraic variety X to the study of algebraic diffe

From playlist Felix Klein Lectures 2022

Related pages

Banach space | Fréchet derivative | Mathematics | Function (mathematics) | Gateaux derivative | Derivative | Chain rule | Continuous function | Open set