Subspace of a Normed Linear Space
Let $V$ be a closed subspace of a normed linear space $(X, \|\cdot\|_X)$.
1) Show that for every $u_0 \in X\backslash V$, there exists $T \in X^*$ such that $\|T\| = 1$, $T \equiv 0$ on $V$ and $T(\lambda u_0) = \lambda d_X(u_0, V)$ for all $\lambda \in \mathbb{R}$, where $d_X(u_0, V) = \inf\{\|u-u_0\|_X: u \in V\}$.
2) Show that $d_X(u_0, V) = \max\{T(u_0): T \in X^*, \|T\| = 1, T \equiv 0$ on $V\}$
Malindad
53
The answer is accepted.
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