What does an infinite dimensional vector space even mean? What would be some good intuitive examples?

Question

I am looking for some examples other than from quantum mechanics, if possible. Simple examples.

Answer 1

except for quantum mechanics there are many other physical examples and applications for infinite dimensional vector spaces and manifolds especially in the context of so called "field theories". The most intuitive and visual ones in my opinion are a temperature field and a water surface or guitar string:

Imagine a calm lake with no waves or disturbance at first. The surface is described by a continuum of "infinitely" many water molecules in equilibrium (they don't move on their own until there is a disturbance). Now take a continuous function $\phi(x)$ which tells you the vertical displacement of the molecule at (surface) position x ($\in R^2$ if you view the lake from above as a real plane).

Now the possible spatial configuration of this system is given by a vector space of functions. Instead of a few coordinates to describe the position of a point particle you now need a whole continuous or smooth function to describe the position of every point of the surface. So your configuration space is infinite dimensional. The guitar string would be an easier analogue of this idea.

There are some more examples for example in Abraham and Marsden's book Foundations of Mechanics (chapter infinite dimensional Hamiltonian systems), but I still find them too difficult to give intuition.

I hope this helps a bit :)