Above is a theorem in p.74 of "Functional Analysis" by Peter Lax.
Above is theorem 8 of chapter 3.
My question : Let $X$ be a normed linear space over the real or complex numbers and $Y$ be a subspace, $l$ be a bounded linear functional on $Y$. If $l:Y\to\mathbb{C}$ has operator norm $C$, then can it be extended to $X$ so that the extension has the same operator norm as $l$?