Question

If f(x) = f(y) for every bounded linear functional f on a normed space X, show that x = y.

Answer 1

Answer on Question #83090 – Math – Functional Analysis

Question

If $f(x) = f(y)$ for every bounded linear functional $f$ on a normed space $X$, show that $x = y$.

Solution

It is false.

Let $f(x)$ be different from 0 bounded linear functional $f$, write its kernel:

Ker $f = \{x \in X : f(x) = 0\} \neq 0$ and take $x_0 \in \text{Ker } f$, $x_0 \neq 0$ (such an element $x_0$ always exists) then for elements $x + x_0$ and $x$ we have $f(x + x_0) = f(x) + f(x_0) = f(x)$ but $x \neq x + x_0$.

Answer: It is false.

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