Kenneth Kuttler

12.1. BASIC INEQUALITIES AND PROPERTIES 359

b.) (∫|a f |p dµ)1/p = |a| (

∫| f |p dµ)1/p if a is a scalar.

c.) (∫| f +g|p dµ)1/p ≤ (

∫| f |p dµ)1/p +(

∫|g|p dµ)1/p.

f → (∫| f |p dµ)1/p would define a norm if (

∫| f |p dµ)1/p = 0 implied f = 0. Unfor-

tunately, this is not so because if f = 0 a.e. but is nonzero on a set of measure zero,(∫| f |p dµ)1/p = 0 and this is not allowed. However, all the other properties of a norm are

available and so a little thing like a set of measure zero will not prevent the considerationof Lp as a normed vector space if two functions in Lp which differ only on a set of measurezero are considered the same. That is, an element of Lp is really an equivalence class offunctions where two functions are equivalent if they are equal a.e. With this convention,here is a definition.

Definition 12.1.7 Let f ∈ Lp (Ω). Define ∥ f∥p ≡ ∥ f∥Lp ≡ (∫| f |p dµ)1/p .

Then with this definition and using the convention that elements in Lp are considered tobe the same if they differ only on a set of measure zero, ∥ ∥p is a norm on Lp (Ω) becauseif || f ||p = 0 then f = 0 a.e. and so f is considered to be the zero function because it differsfrom 0 only on a set of measure zero.

The following is an important definition.

Definition 12.1.8 A complete normed linear space is called a Banach1 space.

Lp is a Banach space. This is the next big theorem which says that these Lp spaces arealways complete.

Theorem 12.1.9 The following holds for Lp(Ω,F ,µ), p ≥ 1. If { fn} is a Cauchysequence in Lp(Ω), then there exists f ∈ Lp (Ω) and a subsequence which converges a.e.to f ∈ Lp(Ω), and ∥ fn− f∥p→ 0.

Proof: Let { fn} be a Cauchy sequence in Lp(Ω). This means that for every ε > 0 thereexists N such that if n,m≥ N, then ∥ fn− fm∥p < ε . Now select a subsequence as follows.Let n1 be such that ∥ fn− fm∥p < 2−1 whenever n,m≥ n1. Let n2 be such that n2 > n1 and∥ fn− fm∥p < 2−2 whenever n,m ≥ n2. If n1, · · · ,nk have been chosen, let nk+1 > nk andwhenever n,m ≥ nk+1,∥ fn− fm∥p < 2−(k+1). The subsequence just mentioned is { fnk}.Thus

({ω :∣∣ fnk (ω)− fnk+1 (ω)

∣∣p > (23

)k})≡ µ (Ek)

≤(

)k ∫Ek

∣∣ fnk (ω)− fnk+1 (ω)∣∣p dµ

1These spaces are named after Stefan Banach, 1892-1945. Banach spaces are the basic item of study in thesubject of functional analysis and will be considered later in this book.

There is a recent biography of Banach, R. Katuża, The Life of Stefan Banach, (A. Kostant and W. Woyczyński,translators and editors) Birkhauser, Boston (1996). More information on Banach can also be found in a recentshort article written by Douglas Henderson who is in the department of chemistry and biochemistry at BYU.

Banach was born in Austria, worked in Poland and died in the Ukraine but never moved. This is becauseborders kept changing. There is a rumor that he died in a German concentration camp which is apparently nottrue. It seems he died after the war of lung cancer.

He was an interesting character. He hated taking examinations so much that he did not receive his undergraduateuniversity degree. Nevertheless, he did become a professor of mathematics due to his important research. He andsome friends would meet in a cafe called the Scottish cafe where they wrote on the marble table tops until Banach’swife supplied them with a notebook which became the ”Scotish notebook” and was eventually published.

12.1. BASIC INEQUALITIES AND PROPERTIES 359b.) (af |? dp)!” = |a| (ffl? dp)!” if a is a scalar.c) (f|ft slau!” < (S| flPau)'"? +(f\gi?au)'””.f— (f|fl?du)'/? would define a norm if (['|f|’dw)!/? =0 implied f = 0. Unfor-tunately, this is not so because if f = 0 a.e. but is nonzero on a set of measure zero,(f\fl? dp) '/P _ 9 and this is not allowed. However, all the other properties of a norm areavailable and so a little thing like a set of measure zero will not prevent the considerationof L? as a normed vector space if two functions in L? which differ only on a set of measurezero are considered the same. That is, an element of L’ is really an equivalence class offunctions where two functions are equivalent if they are equal a.e. With this convention,here is a definition.Definition 12.1.7 Ler f <1? (Q). Define ||f\|, = fll» = Ifa)”.Then with this definition and using the convention that elements in L? are considered tobe the same if they differ only on a set of measure zero, || ||, is a norm on L? (Q) becauseif ||f||, =0 then f = 0 a.e. and so f is considered to be the zero function because it differsfrom 0 only on a set of measure zero.The following is an important definition.Definition 12.1.8 4 complete normed linear space is called a Banach! space.L? is a Banach space. This is the next big theorem which says that these L? spaces arealways complete.Theorem 12.1.9 The following holds for L?(Q,F,U),p > 1. If {fn} is a Cauchysequence in L?(Q), then there exists f € L? (Q) and a subsequence which converges a.e.to f € L?(Q), and || fn — f |p > 9.Proof: Let {f,} be a Cauchy sequence in L?(Q). This means that for every € > 0 thereexists N such that if n,m > N, then || fn — fml|p < €. Now select a subsequence as follows.Let n; be such that || fn — fin|lp < 2-! whenever n,m > nj. Let nz be such that ny > n; and\|fn — Fmllp < 2-* whenever n,m > nz. If nj,--+ ,ng have been chosen, let ng; > ng andwhenever n,m > ng+1,||fn —fin\lp < 2~“+. The subsequence just mentioned is { fy, }-Thusk(40: |fu (0) fn. (@)|” > (5) p ) =e (Ei< 3) / [fs (@) —Fin, (|'These spaces are named after Stefan Banach, 1892-1945. Banach spaces are the basic item of study in thesubject of functional analysis and will be considered later in this book.There is a recent biography of Banach, R. Katuza, The Life of Stefan Banach, (A. Kostant and W. Woyczyiski,translators and editors) Birkhauser, Boston (1996). More information on Banach can also be found in a recentshort article written by Douglas Henderson who is in the department of chemistry and biochemistry at BYU.Banach was born in Austria, worked in Poland and died in the Ukraine but never moved. This is becauseborders kept changing. There is a rumor that he died in a German concentration camp which is apparently nottrue. It seems he died after the war of lung cancer.He was an interesting character. He hated taking examinations so much that he did not receive his undergraduateuniversity degree. Nevertheless, he did become a professor of mathematics due to his important research. He andsome friends would meet in a cafe called the Scottish cafe where they wrote on the marble table tops until Banach’swife supplied them with a notebook which became the ’’Scotish notebook” and was eventually published.