Equivalence Between Definitions of Adjoint Operator I have encountered two different definitions for the adjoint of a linear operator, and I can't seem to see how they are equivalent.
The first definition is as follows:
Let $(V,\|\cdot\|_V)$, and $(W, \|\cdot\|_W)$ be normed vector spaces.
Define the dual of $V$ as $V^*=\left\{\phi:V\rightarrow \mathbb{C}| \phi\text{ is a bounded linear  transformation}\right\}$.
The adjoint of $T$ is the linear transformation:
$$T^*:W^*\rightarrow V^* \text{, given by } (T^*\phi)\,(v)=\phi(T(v))$$
The second defintion is as follows:
Let $(H_1,\langle\cdot\rangle_1)$, and $(H_2,\langle\cdot\rangle_2)$, be Hilbert Spaces and let $T:H_1\rightarrow H_2$ be a bounded linear transformation.
The adjoint of T is the function $T^*:H_2\rightarrow H_1$, given by $\langle Tx,y\rangle_2=\langle x,T^*y\rangle_1$
I know the first definition is more general, it applies to any normed vector space, while the second only applies to Hilbert Spaces. This makes me think that somehow, the first definition can be specialized to the the scond for the case of a Hilbet Space. I've been trying this for a while, and I think it involves the Riesz Representation Theorem, but I'm not quite sure how to go about it.
The Riesz Representation Theorem says, that for a Hilbert Space $(H,\langle\cdot\rangle)$:
$\Psi:H\rightarrow H^*$,  $x \mapsto \Psi_x$, where:
$\Psi_x:H\rightarrow \mathbb{C}$, $h\mapsto\langle h,x\rangle$,
Is a skew linear bijective isometry with operator norm of $1$.
If anyone could offer some help on how to get from the first definition to the second, I'd be much appreciative, and much less confused about this whole business!
 A: The Riesz theorem gives you an anti-linear isometry between $H$ and $H^*$. Lets call it $i: H\to H^*$ and its inverse $i^*:H^*\to H$. You have that $i(x)\ (y) = \langle x,y\rangle$. The inverse is actually the adjoint of the map in the topological vectorspace sense, which is why I used the notation $i^*$.
So let $T:H_1\to H_2$ be continuous linear. The topological vectorspace adjoint is a map $T^*: H_2^*\to H_1^*$. Using our anti-linear isometries $i_1^*, i_2$ we can get a map
$$i_1^*\circ T^*\circ i_2: H_2\to H_2^*\to H_1^*\to H_1$$
as a composition of two anti-linears and one linear, this is a linear map. By putting in the definitions you can explicitly check that it satisfies the defining equation of the Hilbert space adjoint:
$$\langle i_1^*T^*i_2 x,y\rangle_1 =(i_1i_1^*T^*i_2)(x)\,(y)= T^*i_2(x)\,(y)=i_2(x)\,(Ty)=\langle x,Ty\rangle_2$$
Alternatively you can look at $i_2^* \circ T\circ i_1: H_1^*\to H_2^*$ and take the adjoint of that, but its the same as the above:
$$(i_2^* \circ T\circ i_1)^*=i_1^*\circ T^*\circ i_2$$
