Re: infinite vector spaces

{1, x, x^2, x^3, ..., x^n, ....} is an infinite subset of the set of
real-valued functions, and the vectors in this subset are linearly
independent (e.g., by the Fundamental Theorem of Algebra). Thus, there
cannot be a finite basis.


vsgdp wrote:
In general, how does one prove that a vector space is infinite dimensional?
My first attempt would be to suppose it were finite, then there exists a
finite basis, and then show that there is some vector in the vector space
that cannot be written as a linear combination of those finite basis
vectors. But how would you do this for say the set of all real-valued
functions, for example?I've only learned the trigonometric bases for
functions from Fourier series, so it is difficult for me to make progress
with my idea.

Dr. Michael W. Ecker
Associate Professor of Mathematics
Pennsylvania State University
Wilkes-Barre Campus
Lehman, PA 18627

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Relevant Pages

  • Re: infinite vector spaces,
    ... finite basis, and then show that there is some vector in the vector space ... that cannot be written as a linear combination of those finite basis ... if a space has a known infinite-dimensional subspace then it is ...
    (sci.math)
  • Re: infinite vector spaces
    ... real-valued functions, and the vectors in this subset are linearly ... by the Fundamental Theorem of Algebra). ... cannot be a finite basis. ...
    (sci.math)
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