I am self studying functional analysis and I don't not see the utility of authors trying make it clear that a space $X$ is a Banach space before proceeding with a definition.
For example, going through calculus, I don't recall in our Fourier transform lesson that the professor had to make sure that $X, Y$ were "X" spaces before proceeding with the definition of Fourier transform.
Now I am hesitant to even integrate trivial functions such as $f(x) = x$ because I don't know if integration works on the space that contains $x$.
For example,
Let $X$, $Y$ be Banach spaces, the directional derivative of $f : X → Y$ at $x ∈ U ⊆ X$ in the direction $h ∈ X$, denoted by the symbol $f(x; h)$, is defined by the equation
$$f(x; h) = \lim_{t→0} \frac{f(x + th) − f(x)}{t}$$ whenever the limit on the right exists.
Does it imply that the formula only works between Banach spaces and cannot work on any other space? What if I replaced Banach space in the definition with function space, Hilbert space or simply sets? If so, does that mean I should painstakingly prove that a function is an element of an appropriate functional space before doing even the simplest of calculations?
Why do authors insist on making clear that a space is a Banach space before proceeding with a definition? How would the definition fail if $X, Y$ were not Banach space?
A Banach space is a complete normed space. If it is not complete, you loose a lot of things. Just like you loose things if you don't work with $\mathbb{R}$, and work only with $\mathbb{Q}$ (since not every Cauchy sequence in $\mathbb{Q}$ converge to a point in $\mathbb{Q}$...but every Cauchy sequence in $\mathbb{R}$ converge to a point in $\mathbb{R}$; that's why $\mathbb{R}$ is called a complete metric space).