Geometry of the space $C[a, b]$ respect to the norm $\lVert x \rVert_{\infty} = \max_{t\in [a,b]}\lvert x(t)\rvert$.

Question

I have studied that the space $C[a, b]$ of all scalar-valued (real or complex) continuous functions defined on $[a, b]$ is a Banach space with respect to the norm $\lVert x \rVert_{\infty} = \max_{t\in [a, b]}\lvert x(t)\rvert$.

I want to understand geometry of the space $C[a, b]$ for the norm defined above. Is this possible? Because I am not able to visualize this space with respect to the norm defined above. Could anybody explain me?

Thanks

Answer 1

This questions is rather broad. As soon as you go from the finite dimensional space to the infinite dimensional visualizing becomes much more complicated in general. One needs to develop certain intuition which comes from practice. There is nothing special about $C[a,b],$ you have to get used to infinite dimensional spaces first. When you are dealing with concrete problem you can try to apply your "euclidean" intuition first. Roughly speaking, there are two major differences to keep in mind:

$1.$ There is no Pythagorean theorem $\|x\|^2+\|y\|^2=\|x+y\|^2$ whenever one $x,y$ are "orthogonal." The reason is that we cannot define orthogonality in the usual way because $C[a,b]$ is not a Hilbert space. One can however, choose "almost" orthogonal vectors if necessary.

$2.$ The unit ball is not compact, but this is a common thing for all infinite dimensional spaces. One has to be very careful when choosing convergent subsequences etc.

Answer 2

All you can "visualize" in this space is the convergence as the notion of distance is fairly easy to visualize here. So, $f$ and $g$ are close if their graphs are within a certain distance of each other. This is will help if you study the space of $n-$times continuously differentiable functions also.