# Show map is injective

1. Mar 3, 2008

### quasar987

[SOLVED] Show map is injective

1. The problem statement, all variables and given/known data
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

$$<T(x),y> = \int_Ex(t)y(t)dt$$

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!

Last edited: Mar 3, 2008
2. Mar 4, 2008

### 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).