Stuck on this homework question, would appreciate any help thanks.
Suppose $E$ is a normed space and $E_0$ is a subspace of $E$, all over a field $\mathbb{K}$. Let $T_0 : E_0 \rightarrow l^\infty $ be a bounded linear operator, where as usual $l^\infty$ is the banach space of all bounded sequences in $\mathbb{K}$ with norm $||(x_n)_{n \in \mathbb{N} } || = sup_{n \in \mathbb{N}} |x_n | $. Then use the Hahn–Banach theorem to show that there is an extension of $T_0$ to $E$, call it $T$, where $|| T || = || T_0 || $.
I think this would be really easy if instead of $l^\infty$ it was just $\mathbb{K} $ ( precisely the statement of the hahn-banach theorem) but not sure how this still works when we change to a different normed space.
Best Answer
Let $T_nx$ be the $n-$th coordinate of $Tx$. There exists an extension $S_n$ of $T_n$ to $E$ with $\|T_n\|=\|S_n\|$. Let $Sx=(S_1x,S_2x,...)$. It is fairly easy to check that $S$ an extension with $\|S\|=\|T\|$.