I'm approaching the study of Banach spaces, and I would like to know from you, who know a lot and are actually also goot at explanations, what could be the difference between a function $f: \mathbb{R}\to \mathbb{R}$ definied as $f(x) = x \sin(x)$ (this is merely an example, I could have chosen something else) in the "usual context" and a function $f: \mathbb{R}\to \mathbb{R}$ between banach spaces ($\mathbb{R}$) defined in the same way.
I mean, is there some difference / different property in chosing Banach space in particular, instead of the "usual" $\mathbb{R}$?
Thank you
Yes, the most important difference is that a general Banach space $X$ may be infinite-dimensional. This, for example, means that the unit ball $$ B:=\{x \in X \mid ||x||\leq 1 \} $$ is no longer compact. In the finite-dimensional case $X=\mathbb{R}^n$, the unit ball is compact independent of the norm.
Since you brought up $\mathbb{R}$, you even got more structure on $\mathbb{R}$ than on most vector spaces like $\mathbb{R}^n$. $\mathbb{R}$ is even a field!