Let ( X , A , μ ) be a finite measure space. Fix 1 &lt; p , q &lt; ∞ with 1 p + 1 q = 1. Let { f n } be a sequence in L p ( μ ) such that sup ‖ f n ‖ p = M &lt; ∞. Suppose for every g ∈ L q ( μ ) we have lim n ∫ f n g d μ converges.Prove that f n weakly converges to some f ∈ L p ( μ ).I have shown that ϕ ( g ) = lim n ∫ f n g d μ defines a linear functional ϕ ∈ ( L q ( μ ) ) ∗ . I think f is the element in Lp(μ) corresponding to ϕ ∈ ( L q ( μ ) ) ∗ by the Riezs' theorem isomorphism, but how do I prove this? How do I find this f?

Question

Let   (  X  ,      A    ,  μ  ) be a finite measure space. Fix   1  &amp;lt;  p  ,  q  &amp;lt;  ∞ with       1    p    +      1    q    =  1. Let   {      f    n    } be a sequence in             L      p        (  μ  ) such that   sup  ‖      f    n        ‖    p    =  M  &amp;lt;  ∞. Suppose for every   g  ∈            L      q        (  μ  ) we have       lim    n    ∫      f    n    g    d  μ converges.Prove that       f    n   weakly converges to some   f  ∈      L    p    (  μ  ).I have shown that   ϕ  (  g  )  =      lim    n    ∫      f    n    g    d  μ defines a linear functional   ϕ  ∈  (      L    q    (  μ  )      )    ∗  . I think f is the element in Lp(μ) corresponding to   ϕ  ∈  (      L    q    (  μ  )      )    ∗   by the Riezs&#039; theorem isomorphism, but how do I prove this? How do I find this   f?

Hugo Bruce · Accepted Answer

If   ϕ is a bounded linear functional on       L    q    (  μ  ) then there exists   f  ∈      L    p    (  μ  ) with   ϕ  (  g  )  =      ∫    f    g        d    μ  . You don&#039;t have to find it; Riesz did that for you.Under your definition of   ϕ this means      lim    n    ∫      f    n    g    d  μ  =  ∫  f  g    d  μfor all   g  ∈      L    q    (  μ  ), verifying weak convergence.

Carly Roy · Accepted Answer

Clearly,   ϕ  (  g  )  =  lim      ∫    X        f    n    g    d  μ is well-defined and linear. Moreover,      |    ϕ  (  g  )      |    ≤  M  ‖  g      ‖    q  so   ‖  ϕ  ‖  ≤  M.By Riesz Representation, there exists a unique   f  ∈      L    P   such that   ϕ  (  g  )  =      ∫    X    f  g    d  μ. But this means   lim      ∫    X    (      f    n    −  f  )  g    d  μ  =  0.

Let ( X , <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvaria

Answered question

Answer & Explanation

New Questions in Pre-Algebra