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Posts Tagged ‘Hahn-Banach’

Separable space 2

January 4, 2011 2 comments

Denote c_0=\{(\alpha_n) \subset \Bbb{C} : \alpha_n \to 0\}, the set of complex sequences which converge to 0. Furthermore, consider the sequences x_n=((n+1)^{-j})_j. Prove that the closed linear span of \{x_n : n=1,2,..\} is in fact c_0.

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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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Hahn-Banach application.

January 19, 2010 Leave a comment

Suppose X is a normed space and X_0 is a closed subspace of X and x_0 \in X \setminus X_0. Then we can find f \in X^\prime such that f(x_0)=1 and f(x)=0,\ \forall x \in X_0.

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Hahn-Banach application. The dual is not trivial

January 19, 2010 Leave a comment

Denote by X^\prime=\{ f: X \to \mathbb{K} : f is linear and continuous \} where X is a Banach space over \mathbb{K}. Prove that X^\prime \neq \{0\}, in fact, for every x \neq 0 \in X, we can find f \in X^\prime such that f(x)=\|x\| and \|f\|=1. Read more…

Hahn-Banach (complex version)

January 19, 2010 Leave a comment

Let X be a complex vector space, X_0 one of its subspaces, p: X \to \mathbb{R}_+ such that p(\lambda x)=|\lambda| p(x),\ \forall \lambda \in \mathbb{C}, x \in X and p(x+y) \leq p(x)+p(y),\ \forall x,y \in X, satisfying |f(x)| \leq p(x),\ \forall x \in X_0, where f:X_0 \to \mathbb{C} is linear.
Under these conditions, there exists a linear functional F :X \to \mathbb{C} such that F| _{X_0}=f and |F(x)| \leq p(x),\ \forall x \in X. Read more…

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Hahn-Banach Theorem (real version)

January 18, 2010 Leave a comment

Suppose X is a vector space over \mathbb{R}, p:\mathbb{X} \to \mathbb{R} has the following properties: p(\lambda x)=\lambda p(x),\ \forall x \in X, \lambda \in\mathbb{R}_+ and p(x+y)\leq p(x)+p(y),\ \forall x,y \in X.
Let X_0 be a subspace of X and u: X_0\to \mathbb{R} a linear functional such that u(x) \leq p(x),\ \forall x \in X_0.
Then we can find f:X \to \mathbb{R} a linear functional such that f| _{X_0} =u and f(x) \leq u(x),\ \forall x \in X. Read more…

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