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

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with induced norm $\lVert\cdot\rVert$, which is a norm on $V$ by claim 2 of The Induced Norm is a Norm, and Induces a Metric. Let $n$ be a natural number, let $e\in V^{n}$ be an $n$-tuple in $V$ that is an orthonormal basis of $V$, with components $e_{k}$, and let $T$ be a linear operator on $V$.

Then $T$ is a bounded linear operator; more precisely, with the finite sum of real numbers

$$C=\sum_{k=1}^{n}\bigl\lVert T(e_{k})\bigr\rVert ,$$

one has $0\le C$ and

$$\lVert T(u)\rVert\le C\,\lVert u\rVert\qquad\text{for every }u\in V,$$

the order being that of the ordered field of real numbers.