Re: infinite vector spaces,
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Sat, 7 Oct 2006 19:43:47 +0000 (UTC)
In article <fDRVg.673$zf3.307@fed1read03>, vsgdp <hello@xxxxxxxx> wrote:
In general, how does one prove that a vector space is infinite dimensional?
(i) You can exhibit an infinite linearly independent set. For example,
you can prove that the vector space of polynomials is infinite
dimensional by noting that the set {1, x, x^2, x^3, ..., x^n, ...} is
linearly independent.
(ii) You can prove that no finite set spans the vector space. For
example, again in the vector space of polynomials, let {p_1,...,p_m}
be any finite set of polynomials. Let N be the maximal degree of the
p_i. Then x^{N+1} is not in span(p_1,...,p_m), so {p_1,...,p_m} does
not span.
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?
For this, I would suggest method (i) above.
--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
.
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