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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.

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    Do you know anything about Fourier series?2011-07-19
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    there are different notions of "basis". the algebraic one (sometimes called a hamel basis) is a collection of independent vectors st every vector can be written as a _finite_ linear combination of basis elements. in something like $L^2(S^1)$ you might consider the orthonormal basis $\{\cos(nx), \sin(nx) : n=0,1,2,3,...\}$ where $L^2$ functions can be written as _infinite_ linear combinations (fourier series) of the basis functions.2011-07-19
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    @Theo Buehler: Hm?2011-07-19
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    @Theo But in $f(x)=\sin(x)$, say $x=1$ (basis=1), then $f(x)=\sin(1)$ which is a value, not a function2011-07-19
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    @Christian Blatter: Oh... that was a major lapse. (How could 4 people agree?) `@bobobobo: Sorry about that. GleasSpty and Agustí expand on what I was trying to say, but correctly.2011-07-19
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    Can anyone offer an answer to the question _what does it mean to be infinite dimensional_, addressing the idea that __3d space allows you to move in 3 distinct directions__, so how does a function represent infinite-dimensional space, allowing you to move in ___"infinitely many directions?"___2011-07-19
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    @bobobobo: Is the answer I posted satisfactory? In short, you can move along any of infinitely many basis vectors, but to get to any given point, you only need to move along some finite number of them.2011-07-21

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