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A new tag created by Rebellos, they also created the tag-excerpt and the tag-wiki.
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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 $(X,\|\cdot\|)... 4 Exercise : Let$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 \|$. Thoughts : I apologise for not providing a proper attempt but this is one of the first such exercises I'm handling, so I seem at loss. Initially, ... 1 Exercise : Let$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...

In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry. The theorem is named for the mathematicians Hans Hahn and Stefan Banach...