If $u_n\to u$ in $L^p$ and $u_n\to v$ in $L^q$ do we have that $u=v$?

Question

If $u_n\to u$ in $L^p$ and $u_n\to v$ in $L^q$ do we have that $u=v$ ?

Attempt

I know that we have a subsequence $u_{n_k}$ that converge to $u$ and $v$ a.e. and thus that $u=v$ a.e., but do we have that $u=v$ everywhere ?

The conditions given do not specify u and v everywhere, but you can choose u and v to be the same representative if you want. — Ian, Oct 06 '17 at 15:58

score 0 · Answer 1 · answered Oct 20 '18 at 14:42

0

The argument is correct. Elements of $L^p$ are equivalence classes for the relation of equality almost everywhere. Therefore the equality $u=v$ means that $u=v$ almost everywhere.

answered Oct 20 '18 at 14:42

Davide Giraudo

172,925

If $u_n\to u$ in $L^p$ and $u_n\to v$ in $L^q$ do we have that $u=v$?

1 Answers1