I have been reading Kreyszig's Functional Analysis when I encountered two versions of Riesz's Representation Theorems:
(1) Every bounded linear functional $f$ on a Hilbert space $H$ can be represented in terms of the inner product, namely $f(x)=\langle x,z\rangle$, where $z$ is uniquely determined by $f$ and $\|z\|=\|f\|$.
(2) Let $H_{1},H_{2}$ be Hilbert spaces and $h:H_{1}\times{}H_{2}\rightarrow{}K$ a bounded sesquilinear form. Then $h$ has a representation $h(x,y)= \langle Sx,y \rangle$, where $S:H_{1}\rightarrow{}H_{2}$ is a bounded linear operator. $S$ is uniquely determined by $h$ and has norm $\|S\|=\|h\|$.
The book used (1) to deduce (2), but I am curious whether you can deduce (1) given (2)? Is there a way to understand (1) as a special case of (2)?
Also, I find these two theorems very similar to each other. So I'm curious if is there a deeper connection between them, other than the superficial similarity in their names and their mutual concern for norm equality?
Sorry if this is a vague questions...
Thank you!!