This question is not about matrices.
Kunze, Linear algebra (1971) defines transpose of a linear transformation and adjoint of a linear transformation.
Kunze states the relation between the adjoint of $T$ and the transpose of the matrix of $T$, but I could not find the relation between the adjoint of $T$ and the transpose of $T$ as a linear transformation.
Transpose:
Adjoint:
What is the relation between these two definitions? Note again, I am not asking about the relation between the adjoint of $T$ and the transpose of the matrix of $T$, so questions like this one are not duplicates.
Best Answer
Let $\textsf V$ and $\textsf W$ be vector spaces over $F$ with the inner products $\langle \cdot,\cdot \rangle_1$ and $\langle \cdot,\cdot \rangle_2$, respectively.
For $v\in\textsf V$, define the linear map $\varphi_1(v) : \textsf V \to F$ by $\varphi_1(v)(u) = \langle u,v \rangle_1$.
It is easy to see that $\varphi_1$ is an injective linear transformation between $\textsf V$ and $\textsf V^*$. Also, since $\textsf V$ and $\textsf V^*$ has the same dimension, $\varphi_1$ is an isomorphism.
Then, we can identify $\textsf V$ with $\textsf V^*$ through $\varphi_1$. The same happens between $\textsf W$ and $\textsf W^*$ with the isomorphism $\varphi_2 : \textsf W \to \textsf W^*$ defined in a similar way.
Now, this maps will help us to identify $\textsf T^*$ with $\textsf T^t$. More precisely, we have the following relationship:
$$\varphi_1 \circ \textsf T^* = \textsf T^t \circ \varphi_2.$$
Note : Here, $\textsf T^*$ is the unique linear transformation, from $\textsf W$ to $\textsf V$, such that
$$\langle \textsf T(x), y \rangle_2 = \langle x,\textsf T^*(y) \rangle_1$$
for all $x\in\textsf V$ and $y\in\textsf W$.