Re: infinite vector spaces,

In article <1160529744.545111.90580@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<schoenfeld.one@xxxxxxxxx> wrote:

Shmuel (Seymour J.) Metz wrote:
In <fDRVg.673$zf3.307@fed1read03>, on 10/07/2006
at 10:44 AM, "vsgdp" <hello@xxxxxxxx> said:

In general, how does one prove that a vector space is infinite
dimensional?

Find an infinite set of linearly independent vectors.

Or rather, an infinite linearly independent set of vectors.

Linear independence is not a sufficient condition for orthogonality.

So what? That's not what you asked for. On the other hand,
given an infinite linearly independent set (and an inner
product) you can get an infinite orthonormal set by
Gram-Schmidt.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

  • Re: infinite vector spaces,
    ... Robert Israel wrote: ... an infinite linearly independent set of vectors. ...
    (sci.math)
  • Re: infinite vector spaces,
    ... an infinite linearly independent set of vectors. ... Orthogonality is an unnecessary condition when choosing set of basis ...
    (sci.math)
  • Re: infinite vector spaces,
    ... Shmuel Metz wrote: ... an infinite linearly independent set of vectors. ...
    (sci.math)
  • Re: Linear Algebra
    ... >to span C^3 and they have to be linearly independent. ... linearly independent vectors in a space of dimension k will ... together with the other two vectors, a linearly independent set. ... then a cannot be equal to 0; the second equation tells us that ...
    (sci.math)