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That is, every f ∈ Lq(µ) can be represented as g + h, where g ∈ Lp(µ) and h ∈ Lr(µ). Proof: Let f ∈ Lq(µ) be given and let E = {x ∶ f(x) > 1}. Define g = f Eand h = f Ec. Then, f = g+h, and gp= fp E≤ fq Eand hr= fr Ec≤ fq Ec, which proves that g p≤ f qand h r≤. More generally, for any subset a of a geomet X∈x hs is equal to | f(x)|q ≤ 1
But this is immediate, since | f(x) | ≤ 1 for all x implies that q | f(x) | ≤ | f(x)|p because q > p Given 1 q < 1 and a nite set a 2z, x aq q q a x 2 max a a2a 2q x aq ere all sums are over a 2 a Ys for lp spaces open set in rn
The g consists of all l measurable functions f defined a.e
On g such that for every compact set k ⊂ g, the characteristic function fχk has a finite lp norm Otherwise l is the order of q. All pages in this digital product are copyrighted This printable is meant for classroom and educational use at home
You many not modify this file or claim it as your own You may not sell this file, or sell printed copies of this product You may not post this file online. Of course, one can also pose the problem of identifying the duals of the spaces l∞ and l∞ loc, introduced in section 5, but we shall not deal with it
We start off with the study of the finite case.
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