Proof of Action of an Operator with an Orthonormal Eigenbasis

Claim 1. Let $x\in V$. Since $e$ is an orthonormal basis of $V$, claim 1 of Orthonormal Expansion and Parseval's Identity in Finite Dimensions gives

A linear operator on $V$ is a linear map from $V$ to $V$, so claim 4 of Properties of Finite Sums of Vectors applies to $T$ and yields

Fix $k\in[n]$ and write $c=\langle e_{k},x\rangle$. Condition 2 of Linear Map gives $T(ce_{k})=c\,T(e_{k})=c(\lambda_{k}e_{k})$, condition 5 of Vector Space over a Field gives $c(\lambda_{k}e_{k})=(c\lambda_{k})e_{k}$, and $c\lambda_{k}=\lambda_{k}c$ because multiplication in the field $\mathbb{C}$ is commutative. Hence

and substituting into the previous display proves claim 1.

Claim 2. The operator $T'$ satisfies the same hypotheses as $T$, namely it is a linear operator on $V$ with $T'(e_{k})=\lambda_{k}e_{k}$ for every $k\in[n]$. Hence claim 1, applied to $T'$, gives