Category Archives: Functional analysis

Weighted isoperimetric inequalities in warped product manifolds

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

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Faber-Krahn inequality

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

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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

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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

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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