Properties of the Operator Norm

Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$, let $S$ and $T$ be bounded linear operators on $V$, and let $\lambda$ be a complex number with modulus $|\lambda|$. Write $\lVert S\rVert_{\mathrm{op}}$ and $\lVert T\rVert_{\mathrm{op}}$ for their operator norms, which exist and are unique by Existence and Uniqueness of the Operator Norm; 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 that definition, and the order is that of the ordered field of real numbers. Then the following hold.

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

$$\lVert T(u)\rVert\le\lVert T\rVert_{\mathrm{op}}\,\lVert u\rVert\qquad\text{for every }u\in V.$$

2. (Least admissible constant) If $C$ is a bound for $T$, then $\lVert T\rVert_{\mathrm{op}}\le C$.

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

$$\lVert S+T\rVert_{\mathrm{op}}\le\lVert S\rVert_{\mathrm{op}}+\lVert T\rVert_{\mathrm{op}},\qquad \lVert\lambda T\rVert_{\mathrm{op}}=|\lambda|\,\lVert T\rVert_{\mathrm{op}},\qquad \lVert ST\rVert_{\mathrm{op}}\le\lVert S\rVert_{\mathrm{op}}\,\lVert T\rVert_{\mathrm{op}} .$$