Let $(E, \| \cdot\|)$ be a normed vector space and $(E^*, \| \cdot\|_*)$ its dual space. Given $f \in E^*$ and $x\in E$, we write $\langle f,x \rangle$ instead of $f(x)$. (This is the notation used in my lecture note).
My lecture mentions Corollary 0.1.4:
Here is Riesz theorem:
My thought:
Let $f,g \in F(x_0) \in \mathcal P(E^*)$. Then $\|f\| = \|g\| = \|x_0\|$ and $f(x_0) = g(x_0) = \|x_0\|^2$. By Riesz theorem, there are unique $x_f, x_g \in E$ such that $f(x) = \langle x_f,x \rangle$ and $g(x) = \langle x_g,x \rangle$, and that $\|f\| = \|x_f\|$ and $\|g\| = \|x_g\|$. As such, $x_f =x_g = x_0$ and $f(x)=g(x) = \langle x_0,x \rangle$. Hence I thought that $F(x_0) = \{x \mapsto \langle x_0,x \rangle\}$, but not $\{x_0\}$ as written in the lecture note.
Could you please confirm if my understanding is correct? Thank you so much!
It is customary to identify $H^{*}$ with $H$ in the case of a Hilbert space by identifying $f$ with the unique $x$ in that theorem. So, even though you are right in your description of $F(x_0)$, what is stated in the book is OK with this identification.