Category Archives: Functional analysis

Noncommutative probability I: motivations and examples

Posted on 21/05/2018 by lamwk

In many areas of mathematics, one can usually assign a commutative algebraic structure to the “space” that one is studying. For example, instead of studying an (affine) algebraic variety, one can study algebraic functions on that variety, which gives rise … Continue reading →

Posted in Functional analysis, Probability | Leave a comment

Weighted isoperimetric inequalities in warped product manifolds

Posted on 14/09/2017 by KKK

1. Introduction The classical isoperimetric inequality on the plane states that for a simple closed curve on , we have , where is the length of the curve and is the area of the region enclosed by it. The equality … Continue reading →

Posted in Analysis, Calculus, Differential equations, Functional analysis, Geometry, Inequalities | 2 Comments

Faber-Krahn inequality

Posted on 09/09/2017 by KKK

I record a proof of the Faber-Krahn inequality here, mainly for my own benefit. Let be one of the standard space forms: the Euclidean space , the unit sphere , or the hyperbolic space . Suppose is a bounded domain … Continue reading →

Posted in Analysis, Differential equations, Functional analysis, Geometry, Inequalities | Leave a comment

Weak L^1 is not locally convex

Posted on 11/12/2014 by lamwk

Let be the Lebesgue measure on . Consider , the space of Lebesgue measurable functions for which there exists some constant such that for every , . The purpose of this post is to show that this is not a … Continue reading →

Posted in Functional analysis | Leave a comment

Fourier transform of tempered distributions

Posted on 17/09/2012 by HL

This is an extension to my previous post A short note about tempered distributions. Our objective is to discuss the Fourier transform of distributions and give some “little” applications. Before that let us review some basics. We shall use the … Continue reading →

Posted in Analysis, Functional analysis | 1 Comment

A short note about tempered distributions

Posted on 28/05/2012 by HL

Recently I am doing distributions theory, this notes is for filling small gaps and showing some “applications”. I should put some definitions at the top to make this article self-containing. Unfortunately I do not have much time, so I go … Continue reading →

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A chain of double duals

Posted on 24/04/2012 by Ken Leung

Let X be a Banach space. We know that X embeds into its double dual X** as a closed subspace isometrically via defined by and we say that X is reflexive if this embedding is a surjection. One may continue … Continue reading →

Posted in Functional analysis | 4 Comments

Dual norm in R^n

Posted on 19/12/2011 by HL

I shall start posting some homework solutions that readers may be interested. This post discusses one special property about the Euclidean norm. Let be a norm on . By identifying with its usual Euclidean inner product, we can study the … Continue reading →

Posted in Functional analysis, Linear Algebra | 2 Comments

Closed Graph Theorem implies Open Mapping Theorem

Posted on 06/09/2011 by HL

Today Wongting raised the question in the real analysis prelims at UW about proving that the Closed Graph Theorem (CGT) implies the Open Mapping Theorem (OMT). This is a standard exercise in the first course of functional analysis. Let us … Continue reading →

Posted in Analysis, Functional analysis | 11 Comments

Exercises in Real Analysis II

Posted on 31/08/2011 by leonardwong

This post is a sequel to Exercises in Real Analysis. 4. Let be the triangle , and be the restriction of the planar Lebesgue measure on . Suppose that . Prove that . Solution. Assume on the contrary that . … Continue reading →

Posted in Analysis, Functional analysis | 2 Comments

Fuglede’s Theorem in operator theory

Posted on 28/08/2011 by Ken Leung

The following theorem of Fuglede is a classical result in the theory of normal operators. In the proof, one can appreciate how classical analysis are used in solving problems in operator theory through functional calculus.   Let be a complex … Continue reading →

Posted in Analysis, Functional analysis, Operator Theory | Leave a comment