Dense subspace
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- Community Team
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Dense subspace
Let \(\displaystyle{\left(X,||\cdot||\right)}\) be an \(\displaystyle{\mathbb{R}}\) - normed space and \(\displaystyle{Y}\) a
subspace of \(\displaystyle{\left(X,+,\cdot\right)}\) such that, if \(\displaystyle{f\in X^{\star}=\mathbb{B}(X,\mathbb{R})}\) with \(\displaystyle{f|_{Y}=\mathbb{O}}\) , then
\(\displaystyle{f=\mathbb{O}}\) . Prove that \(\displaystyle{Y}\) is dense on \(\displaystyle{\left(X,||\cdot||\right)}\) .
subspace of \(\displaystyle{\left(X,+,\cdot\right)}\) such that, if \(\displaystyle{f\in X^{\star}=\mathbb{B}(X,\mathbb{R})}\) with \(\displaystyle{f|_{Y}=\mathbb{O}}\) , then
\(\displaystyle{f=\mathbb{O}}\) . Prove that \(\displaystyle{Y}\) is dense on \(\displaystyle{\left(X,||\cdot||\right)}\) .
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- Community Team
- Posts: 426
- Joined: Mon Nov 09, 2015 1:52 pm
Re: Dense subspace
Suppose that \(\displaystyle{Y}\) is not dense on \(\displaystyle{\left(X,||\cdot||\right)}\) . Then,
the set \(\displaystyle{\overline{Y}\neq X}\) is a closed subspace of \(\displaystyle{\left(X,||\cdot\right)}\) and
according to \(\displaystyle{\rm{Hahn-Banach}}\) - theorem, there exists \(\displaystyle{f\in X^{\star}}\) such that
\(\displaystyle{f|_{\overline{Y}}=\mathbb{O}}\) and \(\displaystyle{||f||=1}\) . But now,
\(\displaystyle{y\in Y\implies y\in \overline{Y}\implies f(y)=0\implies f_{Y}=\mathbb{O}}\) and according to the
hypothesis, \(\displaystyle{f=\mathbb{O}}\), a contradiction, since \(\displaystyle{||f||=1}\) .
the set \(\displaystyle{\overline{Y}\neq X}\) is a closed subspace of \(\displaystyle{\left(X,||\cdot\right)}\) and
according to \(\displaystyle{\rm{Hahn-Banach}}\) - theorem, there exists \(\displaystyle{f\in X^{\star}}\) such that
\(\displaystyle{f|_{\overline{Y}}=\mathbb{O}}\) and \(\displaystyle{||f||=1}\) . But now,
\(\displaystyle{y\in Y\implies y\in \overline{Y}\implies f(y)=0\implies f_{Y}=\mathbb{O}}\) and according to the
hypothesis, \(\displaystyle{f=\mathbb{O}}\), a contradiction, since \(\displaystyle{||f||=1}\) .
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