Extending this question, page 447 of Gilbert Strang's Algebra book says
What does it mean for a vector to have infinitely many components? There are two different answers, both good:
1) The vector becomes $v = (v_1, v_2, v_3 ... )$
2) The vector becomes a function $f(x)$. It could be $\sin(x)$.
I don't quite see in what sense the function is "infinite dimensional". Is it because a function is continuous, and so represents infinitely many points? The best way I can explain it is:
- 1D space has 1 DOF, so each "vector" takes you on "one trip"
- 2D space has 2 DOF, so by following each component in a 2D (x,y) vector you end up going on "two trips"
- ...
- $\infty$D space has $\infty$ DOF, so each component in an $\infty$D vector takes you on "$\infty$ trips"
How does it ever end then? 3d space has 3 components to travel (x,y,z) to reach a destination point. If we have infinite components to travel on, how do we ever reach a destination point? We should be resolving components against infinite axes and so never reach a final destination point.