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


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




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


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


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


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


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