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Infinite system of linear equations
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From: "Robert Israel" <isr...@math.ubc.ca>
Newsgroups: sci.math
Subject: Re: Infinite system of linear equations
Date: 13 Nov 2006 14:15:19 -0800
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Jules wrote:
> Suppose we have a doubly-indexed sequence {b_(i, j)} of real numbers,
> with i, j positive integers. Suppose also that we have a sequence
> {c_i} of reals. We wish to find a sequence {a_j} of reals so that the
> sum as j goes from 1 to infinity of b_(i, j) * a_j = c_i for each i.
> This is, in some sense, a collection of countably-many linear equations
> in countably-many variables. Are there any conditions on the
> coefficients b_(i, j) that would guarantee existence and/or uniqueness
> of a solution {a_j}? If there were only finitely-many equations and
> variables (the same number of each), then one could simply check that
> the determinant of the coefficient matrix is non-zero. Is there any
> analog of determinant for an "omega-by-omega" matrix?
For functional analysis to make sense of this sort of system you should
specify Banach spaces X and Y for the sequences {a_j} and {c_i}
respectively, such that (Tx)_i = sum_j b_{i,j} x_j gives a bounded
linear
map T from X into Y. You can then ask whether T is injective or
surjective. However, checking these may not be easy in general.
If T is a Fredholm operator of index 0, you have the Fredholm
alternative:
T x = y has a solution x for every y in Y if and only if
the only solution to T x = 0 is x = 0.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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