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Newsgroups: sci.math
From: "Robert Israel" <isr...@math.ubc.ca>
Date: 13 Nov 2006 14:15:19 -0800
Local: Tues, Nov 14 2006 9:15 am
Subject: Re: Infinite system of linear equations
Jules wrote: For functional analysis to make sense of this sort of system you should > 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? 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 You must Sign in before you can post messages.
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