Exercise : Let $X$ be a normed space and $Y$ be a proper closed subspace of $X$. If $x_0 \notin Y$, show that there exists $f \in X^*$ such that : $$\|f\| = \frac{1}{d(x_0,Y)}, \; f(x_0) = 1 \; \text{and} \; f(y) = 0$$ Attempt : I know the following Lemma : Lemma : Let $(X,\|\cdot\|)...
Exercise : Let $X$ be a normed space. Prove that for all $x \in X$ there exists $f \in X^*$, such that $f(x) = \|x\|^2$ and $ \|f\| = \|x \|$. Thoughts : I apologise for not providing a proper attempt but this is one of the first such exercises I'm handling, so I seem at loss. Initially, ...
Exercise : Let $X$ be a normed space and $M \subset X$. Show that an element $x_0 \in X$ belongs to the set $\overline{\langle M \rangle}$ if and only if $f(x_0) = 0$ for all $f \in X^*$ such that $f|_M = 0$. Attempt : First of all, clarifying that $\overline{\langle M \rangle}$ denotes th...
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