Every Linear Operator on a Space with a Finite Orthonormal Basis is Bounded

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} with \reftext{def:inner-product-norm-2026a}{induced norm} $\lVert\cdot\rVert$, which is a \reftext{def:complex-normed-space-2026a}{norm} on $V$ by claim 2 of \ref{lem:inner-product-norm-is-norm-2026a}. Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, let $e:[n]\to V$ be an \reftext{def:orthonormal-basis-2026a}{orthonormal basis} of $V$, and let $T$ be a \reftext{def:linear-operator-2026a}{linear operator} on $V$.

Then $T$ is a \reftext{def:bounded-linear-operator-2026a}{bounded linear operator}; more precisely, with the \reftext{def:finite-sum-field-2026b}{finite sum} of \reftext{def:real-numbers-c54-2026c}{real numbers}

one has $0\le C$ and

the order being that of the \reftext{def:ordered-field-c54-2026b}{ordered field} of real numbers.