Are linear transformations between infinite dimensional vector spaces always differentiable?

Question

In class we saw that every linear transformation is differentiable (since there's always a linear approximation for them) and we also saw that a differentiable function must be continuous, so it must be true that all linear operators are continuous, however, I just read that between infinite dimensional vector spaces this is not necessarily true. I would like to know where's the flaw in my reasoning (I suspect that linear transformations between infinite dimensional vector spaces are not always differentiable).

Answer 1

The differentiation operator is linear but not bounded in $\mathcal{L}^2(\mathbb{R})$ (which is infinite dimensional over $\mathbb{R}$). Hence, it's not continuous as all continuous linear operators in Banach spaces are bounded. To see that it's not bounded, consider what differentiation does to the sequence of functions $$f_n(x) = \sqrt{n}\, e^{-n^2 x^2}$$

Also, it matters how differentiation is defined. For another example which is kind of different, consider conjugation in complex numbers where we view $\mathbb{C}$ as a $2$-dimensional vector space over $\mathbb{R}$. Conjugation is then linear and continuous because $f(a+bi)=a-bi$ but as a map from $\mathbb{C}$ to $\mathbb{C}$ it's not differentiable because it fails to satisfy the Cauchy-Riemann equations.