Uniqueness of the Adjoint, and Existence in Finite Dimensions

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, and let $T$ be a linear operator on $V$. Adjoints are as in that definition, and sums of vectors are finite sums in $V$. Then the following hold.

1. (Uniqueness) $T$ has at most one adjoint. When an adjoint of $T$ exists we write $T^{*}$ for it.

2. (Existence given a finite orthonormal basis) Let $n$ be a natural number and let $e\in V^{n}$ be an $n$-tuple in $V$ that is an orthonormal basis of $V$, with components $e_{k}$. Then the map sending $u\in V$ to

$$\sum_{k=1}^{n}\bigl\langle T(e_{k}),u\bigr\rangle\,e_{k}$$

is a linear operator on $V$ and is an adjoint of $T$. In particular $T$ has an adjoint $T^{*}$, and $T^{*}(u)$ is given by the displayed formula for every $u\in V$; since the condition in the definition of an adjoint does not mention $e$, the same operator arises from every orthonormal basis of $V$. Moreover $T$ and $T^{*}$ are then bounded linear operators, by Every Linear Operator on a Space with a Finite Orthonormal Basis is Bounded.