While doing one homework problem i encounter a difficulty.
Is it true, if X is Banach space equipped with $||.||_1$ and $||.||_2$. Suppose $||x_n-x||_1\to0$ and $||x_n-y||_2\to0$ then x=y? Here X is subspace of $L^p$ and $L^q$ but p and q are not conjugate. $||.||_1$ is $L^p$ norm and $||.||_2$ is $L_q$ is norm
I can show $x=y$ a.e but i don't know how to show x=y.
Any help please....
In general, this is not true. In fact, it is possible to construct an example with $x \ne y$, see the related post https://math.stackexchange.com/a/426499/58577.
However, in your situation, life is a little bit easier as pointed out by @saz. Every sequence converging in $L^*$ with $*=p$ or $*=q$ has a subsequence, which converges pointwise a.e. This can be used to show $x = y$ a.e. and this is $x = y$.