Relation between adjoint and dual space

Question

I'm learning tensors recently. And I found that the notation for dual spaces $V^*$, the star, is the same as the notation of adjoint operators. My definition of adjoint is the adjoint of $T \in L(V,W)$ such that $(x,T^*y)=(Tx,y)$. Since they have the same star *, I'm wondering if there are any hidden connections between them?

Answer 1

Yes, there is a connection. In general, given $T\in L(V, W)$ we can define $T^*\in L(W^*, V^*)$ by the rule $T^*(f)=f\circ T$.

In the special case when $X$ is a finite dimensional inner product space, all the elements of $X^*$ are of the form $f_y(x)=\langle x,y\rangle$ for some $y\in X$. So in this case we can think of linear functionals simply as elements of $X$.

Now assume $V, W$ are some finite dimensional inner product spaces. Given $T\in L(V,W)$ we can as usual define $T^*: W^*\to V^*$. But if we remember the correspondence between $V$ and $V^*$, and between $W$ and $W^*$, we can instead define a map $T^*: W\to V$.

So, let $y\in W$. Then $T^*y$ is an element of $V$, and hence it defines the linear functional $T^*y(x)=\langle x,T^*y\rangle_V$ in $V^*$. On the other hand, $T^*$ has to satisfy $T^*y=y\circ T$, and so this functional in $V^*$ is defined by:

$T^*y(x)=y(T(x))=\langle T(x), y\rangle_W$

So this intuitively explains why $\langle T(x),y\rangle_W=\langle x,T^*y\rangle_V$ has to hold. This is the motivation for your definition.