proof that spaces are complete
Let’s prove completeness for the classical Banach spaces, say where .
Thus it suffices to prove that etc.
It suffices to prove that each absolutely summable series in is summable in to some element in .
by Fatou’s Lemma. Hence is integrable, and is finite for almost all .
For each such that is finite the series is an absolutely summable series of real numbers and so must be summable to a real number . If we set for those where , we have defined a function which is the limit almost everywhere of the partial sums . Hence is measurable. Since , we have . Consequently, is in and we have
Since is integrable and converges to for almost all , we have
by the Lebesgue Convergence Theorem. Thus , whence . Consequently, the series has in the sum .
Royden, H. L. Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.
|Title||proof that spaces are complete|
|Date of creation||2013-03-22 14:40:09|
|Last modified on||2013-03-22 14:40:09|
|Last modified by||Simone (5904)|