In my notes on functional analysis it mentions that $C([0,1]),\ell^p$ and, $\ell^\infty$ are normed vector spaces, and gives some examples of norms that we can define on them. However, it then simply states that these three spaces are infinite-dimensional normed vector spaces.
The only thing mentioned in my notes so far is in relation to finite-dimensional vector spaces, namely, that a vector space is finite-dimensional if it has a finite basis.
My question(s): how is it exactly that one understands these spaces to be infinite-dimensional; what does it mean to say that they are infinite-dimensional and how do they differ from an example of a finite-dimensional vector space, say, $\mathbb R^n$. Going on what I know about finite-dimensional spaces, is it simply then that an infinite-dimensional space has an infinite basis? How would one visualize this? Can anybody show me why the examples I gave above are indeed infinite-dimensional?