Category Archives: Functional analysis

Weak L^1 is not locally convex

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

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

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

Posted in Functional analysis | Leave a comment

A chain of double duals

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

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

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 | 3 Comments

Exercises in Real Analysis II

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