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From: paul jung on 11 Jun 2010 04:35
Let X=[0,1]. Suppose a sequence {f_n} is in L^p and a function f is in
L^p. If f_n converges weakly to f in some L^{p_1} with p_1 less than
p, does {f_n} they converge weakly in L^p?
From: paul jung on 11 Jun 2010 09:25
On Jun 11, 5:35 pm, paul jung <paulj...(a)gmail.com> wrote:
> Let X=[0,1]. Suppose a sequence {f_n} is in L^p and a function f is in
> L^p. If f_n converges weakly to f in some L^{p_1} with p_1 less than
> p, does {f_n} they converge weakly in L^p?

I found a counterexample where f=0, f_n goes weakly to 0 in L^2, f_n
is in L^3, but the L^3 norms diverge.
From: TCL on 11 Jun 2010 10:59
On Jun 11, 4:35 am, paul jung <paulj...(a)gmail.com> wrote:
> Let X=[0,1]. Suppose a sequence {f_n} is in L^p and a function f is in
> L^p. If f_n converges weakly to f in some L^{p_1} with p_1 less than
> p, does {f_n} they converge weakly in L^p?

The answer would be yes if p_1 was greater than p.
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