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Let $(E, | \cdot |)$ be a Banach space and $(E', \| \cdot \|)$ its dual. Assume that $E'$ is strictly convex. Then for each $x\in E$, there is a unique $f_x \in E'$ such that $\|f_x\|=|x|$ and $\langle f_x, x\rangle = |x|^2$. Clearly, $f_{\lambda x} = \lambda f_x$ for all $\lambda\in \mathbb R$ and $x\in E$. Let $x,y\in E$.

Is there any equality/inequality that relates $f_x, f_y$, and $f_{x+y}$?

Analyst
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1 Answers1

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The duality map of a Banach space $E$ is the map that maps each $x \in E$ to the set $J_x=\{f \in E' ; f(x)=||x||^2, ||f||=||x||\}$. As you pointed out, $E$ is smooth iff for each $x\in X, J_x$ is a singleton, say $\{J(x)\}$. In that case, we identify $J_x$ with $J(x)$. Notice that if $E$ is a Hilbert space then $J$ is in fact the identify map (under that identification), and is thus linear. In fact, Hilbert spaces are the only ones with that property, that is, if the duality map of a Banach space is linear, then it is a Hilbert space.

Evangelopoulos Foivos
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  • In a general setting (not necessarily Hilbert space), is there any norm inequality between $f_x, f_y$, and $f_{x+y}$? – Analyst Mar 14 '22 at 07:23
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    Well, you always have $ ||f_{x+y}||=||x+y|| \le ||x||+||y|| = ||f_x|| +||f_y|| $. If for some $x,y,$ you actually have that $||x+y||=||x||+||y||$ then $||f_{x+y}||=||f_x||+||f_y||$. I am not sure if more can be said – Evangelopoulos Foivos Mar 14 '22 at 08:56