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Let $X$ be a normed linear space and let $F$ be a linear functional defined on $X$. Prove that $F$ is unbounded if and only if for each $a \in X$ and each $r > 0,\, \{F(x): \|x-a\| < r \} = \mathbb{R}$.

I am particularly interested in one direction:

If for each $a \in X$ and each $r > 0,\,\{F(x): \|x-a\| < r \} = \mathbb{R}$ , then $F$ is unbounded.

Is the proof of this direction trivial because $\mathbb{R}$ is unbounded hence $F(x)$ is unbounded? Or is there a rigorous way of writing it?

Emptymind
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1 Answers1

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We have (with $a=0$ and $r=1$), that $\{F(x): \|x\| < 1 \} = \mathbb{R}.$

Now suppose that $F$ is bounded. Then there is $c>0$ such that $|F(x)| \le c$ for all $x \in X$ with $||x|| \le 1.$ Hence

$$\{F(x): \|x\| < 1 \} \subseteq [-c,c],$$

a contradiction.

Fred
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