proof that spaces are complete
Let’s prove completeness for the classical Banach spaces, say where .
Since the case is elementary, we may assume . Let be a Cauchy sequence. Define and for define . Then and we see that
Thus it suffices to prove that etc.
It suffices to prove that each absolutely summable series in is summable in to some element in .
Let be a sequence in with , and define functions by setting . From the Minkowski inequality we have
Hence
For each , is an increasing sequence of (extended) real numbers and so must converge to an extended real number . The function so defined is measurable, and, since , we have
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 .
References
Royden, H. L. Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.
Title | proof that spaces are complete |
---|---|
Canonical name | ProofThatLpSpacesAreComplete |
Date of creation | 2013-03-22 14:40:09 |
Last modified on | 2013-03-22 14:40:09 |
Owner | Simone (5904) |
Last modified by | Simone (5904) |
Numerical id | 8 |
Author | Simone (5904) |
Entry type | Proof |
Classification | msc 46B25 |