Thinking about continuous functions as vectors with continuum-many components. Recently, I've been thinking about the vector space $C[a,b]$. I understand there's a lot out there about bases for this vector space, but what I want to try to understand are two things:

*

*Is the dimension of this space the continuum $\mathfrak{c} = 2^{\aleph_0}$, and

*if so, does it make sense to think of a function $f \in C[a,b]$ as a vector with $\mathfrak{c}$-many components, i.e., something of the form $$f = \langle f(a)\cdots \longleftrightarrow \cdots f(b)\rangle,$$
accounting for every function value for each $x \in [a,b]$?

Thanks for your input.
 A: Try to show that $f_t(x) = e^{tx}$ for each $t \in \mathbb{R}$ are linearly independent functions on $[a,b]$. Thus the dimension is at least $2^{\aleph_0}$. However, $|C[a,b]| = 2^{\aleph_0}$ (to prove this you can consider the comment given below your answer!) and so the dimension is at most $2^{\aleph_0}$ and thus equal to it.
In terms of "does it make sense to consider it as that", the answer depends on what you think is reasonable. It's definitely good fun to assign such geometric reasoning to this space and there's nothing inherently wrong in thinking like that - this is formally correct as an idea. However, it's not obvious that this will be that useful as a way of thinking as there are notoriously many phenomena which happen in infinite dimensional spaces and have no analogue in finite dimensional spaces, for example in your space the closed unit ball under the norm $||f|| = sup_{[a,b]} \{f\}$ is closed and bounded but not compact and equally not all linear mappings between infinite dimensional vector spaces have to be continuous etc.
