Results for query "Brezis exercise"

Go Into Exercise

Could I then write:

He went into exercise.However, it might be acceptable in a context where exercise meant a long-term regimen: He went into exercise to improve his poor health.

Applications of the Weak and Weak$^*$ topologies to PDEs?

Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$.The most remarkable result of these topologies is the Banach-Alaoglu-Bourbaki, which asserts that for the weak$^*$ topology (Over $E^*$), the unit ball: $$B_{E^*}=\{f \in E^* | \ ||f|| \leq 1 \}$$ is compact.

on an inequality of Brezis-Lieb

In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) in the paper) states that the $L^2(\Omega)$-norm of $f$ can be estimated by the $L^2(\partial \Omega)$-norm of its trace on $\partial \Omega$ (times a constant only depending on $\Omega$).

Showing $(M^\perp)^\perp=\overline{M}$

I have a question about a step in proving $(M^\perp)^\perp=\overline{M}$ where $M$ is a linear subspace of a normed vector space $E$.And $M^\perp=\{f\in E^*|\langle f,x\rangle =0\}$

This is the proof from Brezis book:

Proof: Assuming the other direction, show $\subset$ direction.

Gelfand triples form Brezis book

Consider $H$ to be Hilbert space and $V$ to be a Banach space.Brezis says: there is a canonical map $T\colon H^*\to V^*$ that is the restriction to $V$ of continuous linear functionals $\varphi$ on $H$, i.