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[SOLVED] Show map is injective
Going crazy over this.
Let 1<p<2 and q>=2 be its conjugate exponent. I want to show that the map T: L^p(E) --> (L^q(E))*: x-->T(x) where
[tex]<T(x),y> = \int_Ex(t)y(t)dt[/tex]
is injective.
This amount to showing that if
[tex]\int_Ex(t)y(t)dt=0[/tex]
for all q-integrable functions y(t), then x(t)=0 (alsmost everywhere)
Should be easy but I've been at this for an hour and I don't see it!
Homework Statement
Going crazy over this.
Let 1<p<2 and q>=2 be its conjugate exponent. I want to show that the map T: L^p(E) --> (L^q(E))*: x-->T(x) where
[tex]<T(x),y> = \int_Ex(t)y(t)dt[/tex]
is injective.
This amount to showing that if
[tex]\int_Ex(t)y(t)dt=0[/tex]
for all q-integrable functions y(t), then x(t)=0 (alsmost everywhere)
Should be easy but I've been at this for an hour and I don't see it!
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