Let $X,Y$ be Banach spaces. A function $f :X \to Y$ is said to be Gâteaux differentiable at $x$ if there exists a bounded linear operator $A : X \to Y$ such that $$\lim_{r \to 0}\frac{\|f(x+rh)-f(x)-rAh\|}{r}=0$$ for every $h \in X.$

I'm trying to show that chain rule fails for Gâteaux differentiability.

To this end, I'm able to construct several examples in $\mathbb R^n.$ However, I'm finding it difficult to construct such examples in a general Banach space.

My attempt:

If $X$ is an infinite dimensional Banach space and $\phi: X \to \mathbb R$ is a discontinuous linear functional, then $f(x)=\phi(x) \|x\|$ is Gâteaux differentiable but not Frechet differentiable at $0$. Motivated by this example, I have been trying to construct examples using discontinuous linear functionals. However, I haven't been successful so far.

I would like to know if there a way to construct a counterexample irrespective of the space. At the least, I would like to see some counterexamples on spaces other than $\mathbb R^n$(domain/codomain of at least one function shouldn't be $\mathbb R^n$).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.