Proving Injectivity of the Map T: L^p(E) --> (L^q(E))* for 1<p<2 and q>=2

[quasar987]

[SOLVED] Show map is injective

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

&lt;T(x),y&gt; = \int_Ex(t)y(t)dt

is injective.

This amount to showing that if

\int_Ex(t)y(t)dt=0

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!

 

[quasar987]

Got it. Turns out that T is a linear isometry and every linear isometry is injective! (If T(y)=0, then ||T(y)|| = ||y|| = 0 ==> y=0).