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3 Spaces Lp 1. In this part we fix a measure space (, A, ) (we may assume that is
 

Summary: 3 Spaces Lp
1. In this part we fix a measure space (, A, ) (we may assume that is
complete), and consider the A -measurable functions on it.
2. For f L1
(, ) set
f 1 = f L1 = f L1(,) =

|f| d.
It follows from the above inequalities that for c C1
f + g 1 f 1 + g 1,
cf 1 = |c| f 1.
With more work we derive that
f 1 = 0 f = 0 - a.e..
Factorising by the subspace of functions equal 0 -a.e., or redefining the
symbol = between to functions, we conclude that L1
(, ) is a normed
vector space.
3. Definition of a normed space from B, Ch.2 (B-2). Prove that spaces in
B-2, Examples 1, parts 1, 2, 4, 5, 9 (cases l1
, l

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics