Sometimes I encounter a book or a research paper where the author insists on working with Banach spaces instead of Hilbert spaces.
I am curious as to what would be a non-trivial difference between these two setups for optimization related applications.
For example, you can still define a Frechet derivative on Banach spaces. That's fine. And the majority of optimization concepts such as strong convexity only involves the norm, and not the inner product. Convergence of a sequence of points $x(k)$ towards the optimum $x^\star$ also involves the norm.
Something trivial that can be done in Hilbert space but not Banach space would be taking the inner product (obvious). But even the inner product can be simply written as the sum of the product of the coordinates of a vector. Hence that can be skipped also.
Is there really any significant difference between workng with Hilbert versus Banach spaces?