proof that Lp spaces are complete


Let’s prove completeness for the classical Banach spaces, say Lp[0,1] where p1.

Since the case p= is elementary, we may assume 1p<. Let [f](Lp)𝐍 be a Cauchy sequenceMathworldPlanetmathPlanetmath. Define [g0]:=[f0] and for n>0 define [gn]:=[fn-fn-1]. Then [n=0Ngn]=[fN] and we see that

n=0gn=n=0fn-fn-1???<.

Thus it suffices to prove that etc.

It suffices to prove that each absolutely summable series in Lp is summable in Lp to some element in Lp.

Let {fn} be a sequenceMathworldPlanetmath in Lp with n=1fn=M<, and define functions gn by setting gn(x)=k=1n|fk(x)|. From the Minkowski inequalityMathworldPlanetmath we have

gnk=1nfkM.

Hence

gnpMp.

For each x, {gn(x)} is an increasing sequence of (extended) real numbers and so must convergePlanetmathPlanetmath to an extended real number g(x). The function g so defined is measurable, and, since gn0, we have

gpMp

by Fatou’s Lemma. Hence gp is integrable, and g(x) is finite for almost all x.

For each x such that g(x) is finite the series k=1fk(x) is an absolutely summable series of real numbers and so must be summable to a real number s(x). If we set s(x)=0 for those x where g(x)=, we have defined a function s which is the limit almost everywhere of the partial sums sn=k=1nfk. Hence s is measurable. Since |sn(x)|g(x), we have |s(x)|g(x). Consequently, s is in Lp and we have

|sn(x)-s(x)|p2p[g(x)]p.

Since 2pgp is integrable and |sn(x)-s(x)|p converges to 0 for almost all x, we have

|sn-s|p0

by the Lebesgue Convergence Theorem. Thus sn-sp0, whence sn-s0. Consequently, the series {fn} has in Lp the sum s.

References

Royden, H. L. Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.

Title proof that Lp spaces are completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath
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