Existence and Uniqueness of the Operator Norm

Let $V$ be a \reftext{def:vector-space-2026a}{complex vector space} equipped with a \reftext{def:complex-normed-space-2026a}{norm} $\lVert\cdot\rVert$, and let $T$ be a \reftext{def:bounded-linear-operator-2026a}{bounded linear operator} on $V$. Let $B_{T}$ denote the set of those \reftext{def:real-numbers-c54-2026c}{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 \reftext{def:ordered-field-c54-2026b}{ordered field} of real numbers.

Then $T$ has exactly one \reftext{def:operator-norm-2026a}{operator norm}, and it is the \reftext{def:upper-bound-supremum-c54-2026b}{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 $V$.