A question about adjoint operators for bounded operators between Hilbert spaces

Question

Currently I'm self studying functional analysis, namely adjoint operators. In the text, the author gives the following construction/definition of adjoint operators for bounded operators between Hilbert spaces:

Construction/definition: In the case of Hilbert spaces for the operator $A\in L(\mathcal{H}_1\to\mathcal{H}_2)^{[1]}$ the adjoint operator $A^*\in L(\mathcal{H}_2\to\mathcal{H}_1)$ is defined via the relation $$ \langle Ax,y \rangle=\langle x,A^*y \rangle, \text{ for }x\in\mathcal{H}_1\text{ and }y\in\mathcal{H}_2. $$

Prior to this construction/definition, the author wrote the following paragraph:

Paragraph: Applying this notation to the pairing between $X$ and $X^*$, we can rewrite the definition of the dual operator $A^*$ of an operator $A$ is the following (symmetric) way: for all $x\in X$ and for all $f\in Y^*$, $$ \langle Ax,f \rangle=\langle x,A^*f \rangle. $$

My question is fairly straightforward: why in the Construction/definition did the author not take $y\in\mathcal{H}^*_2$, but in Paragraph the author required that $f\in Y^*$? I have a feeling the Riesz representation theorem is hiding in here somewhere, but I don't see this construction fully. Any help is appreciated!

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$^{[1]}$ Note here that $L(X\to Y)$ is just notation for the space of all (continuous) bounded operators between normed spaces $X$ and $Y$.

Answer 1

The general definition of the adjoint operator for $T: L(X \to Y)$ is the following:

$T^*: L(Y^* \to X^*)$ is such operator that $\forall f \in Y^*$ $\forall x \in X$ $(T^*f)(x) = f(Tx)$.

In case of a Hilbert space, we can indeed use the Riesz’s representation theorem to define the corresponding Hermitian conjugate operator on the vector space itself. The conjugate operator is just the adjoint operator acting on vectors as if they are the corresponding linear functionals.

Remark: the conjugate operator is usually called the adjoint too, because it doesn’t cause any confusion.