Existence and Uniqueness of the Operator Norm

Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$, and let $T$ be a bounded linear operator on $V$. Let $B_{T}$ denote the set of those real numbers that are of the form $\lVert T(u)\rVert$ for some $u\in V$ with $\lVert u\rVert\le1$, the order being that of the ordered field of real numbers.

Then $T$ has exactly one operator norm, and it is the least upper bound of $B_{T}$. We write $\lVert T\rVert_{\mathrm{op}}$ for this number, the subscript distinguishing it from the norm $\lVert\cdot\rVert$ of vectors in $V$.