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Let $H$ be a Hilbert space.

Could you please give me examples of linear or nonlinear operators $F: H \to H$ such that $$ \limsup\limits_{\|x-y\| \to 0} \|F(x)-F(y)\| = +\infty \quad \forall x,y\in H ? $$

Boris Petrov
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1 Answers1

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If linear, such an operator would be unbounded. Unbounded linear operators defined on a complete normed space do exist, if one takes the axiom of choice. But there are no concrete examples.

A nonlinear operator is easy to produce. Let $(e_\alpha)$ be an orthonormal basis of $H$. Define $$F(x)=\begin{cases} 0 \quad &\text{ if } \mathrm{Re}\langle x,e_1\rangle \notin \mathbb Q \\ q\,e_1 \quad &\text{ if } \mathrm{Re}\langle x,e_1\rangle = \frac{p}{q} \in\mathbb Q \end{cases} $$ where the fraction $p/q$ is written in lowest terms; $p\in\mathbb Z$ and $q$ is a positive integer. Every open set contains points where $F(x)=0$, and points where $F(x)$ has arbitrarily large norm.

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