# A question about adjoint operators for bounded operators between Hilbert spaces

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 $$A\in L(\mathcal{H}_1\to\mathcal{H}_2)^{}$$ 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!

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

• Do you know that a Hilbert space is canonically isomorphic to its dual (exactly by Riesz representation)? Jun 24, 2021 at 0:36

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 $$\forall f \in Y^*$$ $$\forall x \in X$$ $$(T^*f)(x) = f(Tx)$$.