Properties 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} $\lVert\cdot\rVert$, let $S$ and $T$ be \reftext{def:bounded-linear-operator-2026a}{bounded linear operators} on $V$, and let $\lambda$ be a \reftext{def:complex-numbers-2026a}{complex number} with \reftext{def:complex-modulus-2026a}{modulus} $|\lambda|$. Write $\lVert S\rVert_{\mathrm{op}}$ and $\lVert T\rVert_{\mathrm{op}}$ for their \reftext{def:operator-norm-2026a}{operator norms}, which exist and are unique by \ref{lem:operator-norm-existence-uniqueness-2026a}; throughout, $\lVert\cdot\rVert$ without a subscript denotes the norm of a vector of $V$ and $\lVert\cdot\rVert_{\mathrm{op}}$ the operator norm of an operator on $V$. The sum, scalar multiple and product of operators are as in \reftext{def:operator-operations-2026a}{that definition}, and the order is that of the \reftext{def:ordered-field-c54-2026b}{ordered field} of \reftext{def:real-numbers-c54-2026c}{real numbers}. Then the following hold.

\textbf{1. (Fundamental bound)} $0\le\lVert T\rVert_{\mathrm{op}}$, and

\textbf{2. (Least admissible constant)} If $C$ is a \reftext{def:bounded-linear-operator-2026a}{bound} for $T$, then $\lVert T\rVert_{\mathrm{op}}\le C$.

\textbf{3. (Operations)} The operators $S+T$, $\lambda T$ and $ST$ are bounded linear operators on $V$, and