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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3
votes
Accepted
Does the norm on a sequence space have to be monotone?
For $f\in\mathbb R^{\mathbb N}$ define $\|f\|$ to be the infimum of the reals $B$ such that the graph of $f$ is contained in a finite union of lines of the form $Y=mX+c$ with $|m|,|c|\leq B.$ The infi …
10
votes
Accepted
Smoothness of finite-dimensional functional calculus
Yes. The can be derived from the resolvent formalism.
I'll just do the $C^1$ case and leave higher derivatives as an exercise - ask if it's not clear how to generalize. I am basically using formula ( …