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Posts Tagged ‘Banach’

The space of real vector measures is complete

Let $(X,\mathcal{E})$ be a measure space. For a real vector measure $\mu : \mathcal{E} \to \Bbb{R}^m$ we define its total variation $|\mu|$ by

$\displaystyle |\mu(E)|=\sup\left\{ \sum_{k=0}^\infty |\mu(E_k)| : E_k \in \mathcal{E} \text{ pairwise disjoint }, E=\bigcup_{k=0}^\infty E_k \right\}$.

Prove that the space of vector measures endowed with $|\cdot |$ is a Banach space.

A lemma of J. L. Lions

Let $X,Y$ and $Z$ be three Banach spaces with norms $\|\cdot \|_X,\ \|\cdot \|_Y$ and $\|\cdot \|_Z$. Assume that $X \subset Y$ with compact injection and that $Y\subset Z$ with continuous injection. Prove that

$\forall \varepsilon >0 \exists C_\varepsilon \geq 0$ satisfying $\|u\|_Y \leq \varepsilon \|u\|_X+C_\varepsilon \|u\|_Z,\ \forall u \in X$.

Applications:

1. Prove that for every $\varepsilon >0$ there exists $C_\varepsilon \geq 0$ satisfying$\displaystyle \max_{[0,1]}|u| \leq \varepsilon \max_{[0,1]}|u^\prime|+C_\varepsilon\|u\|L^1,\ \forall u \in C^1([0,1])$.
2. Pick $p>1$. Prove that for every $\varepsilon >0$ there exists $C=C(\varepsilon,p)$ such that $\|u\|_{L^\infty(0,1)} \leq \varepsilon \|u^\prime\|_{L^p(0,1)}+C\|u\|_{L^1(0,1)},\ \forall u \in W^{1,p}(0,1)$.

Source: Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011

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Interesting inequality involving Banach spaces and an operator

Let $E,F$ be two Banach spaces with norms $\|\cdot \|_E, \ \|\cdot \|_F$. Let $T \in \mathcal{L}(E,F)$(space of linear bounded operators $T:E \to F$) be such that $R(T)$ is closed and $\dim N(T)< \infty$. Let $| \cdot |$ denote another norm on $E$ which is weaker than $\|\cdot \|_E$, i.e. $|x | \leq M \|x\|_E, \ \forall x \in E$.

Prove that there exists a constant $C$ such that $\|x\|_E \leq C(\|Tx\|_F+|x|),\ \forall x \in E$.

Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Chapter 2

Conditions for an operator to be bounded

February 4, 2011 2 comments

Let $E$ be a Banach space, and let $T : E \to E^*$ be a linear operator. In each of the following cases, prove that $T$ is bounded.
a) $\langle Tx,x \rangle \geq 0,\ \forall x \in E$;
b) $\langle Tx,y \rangle=\langle Ty,x\rangle,\ \forall x,y \in E$.

Categories: Functional Analysis Tags:

Bilinear Continuous Operator

Let $X,Y,Z$ be three Banach spaces and consider a bilinear operator $F:X \times Y \to Z$. Prove that $F$ is continuous if and only if there exists a constant $C$ such that $\|F(x,y)\|_Z \leq C \|x\|_X \|y\|_Y,\ \forall x \in X,\ \forall y \in Y$.

Categories: Analysis, Functional Analysis Tags:

non-Separable space Example 1

Prove that the space $C=\{f: [1,\infty) \to \Bbb{C},\ f \text{ continuous and bounded }\}$ is not separable.
A Banach space $X$ is called separable if it contains a countable dense subset. Here are some interesting facts about separable spaces.
Prove that if $Y$ is a subset of a separable metric space $(X,d)$ then $Y$ is also separable.
If the dual of a normed vector space $X$ is separable, then the space itself $X$ is separable.