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Posts Tagged ‘Banach’

The space of real vector measures is complete

October 14, 2012 Leave a comment

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.

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A lemma of J. L. Lions

October 13, 2011 Leave a comment

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

September 28, 2011 Leave a comment

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

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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

January 4, 2011 Leave a comment

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.
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Categories: Analysis, Functional Analysis Tags:

non-Separable space Example 1

January 4, 2011 Leave a comment

Prove that the space C=\{f: [1,\infty) \to \Bbb{C},\ f \text{ continuous and bounded }\} is not separable.
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Separable spaces

January 4, 2011 8 comments

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.

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