Convergence in $L^p(\Omega)$ Norm

Question

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Let $u,v,v_n\in L^p(\Omega)$ and suppose that $$\|u+v_n\|_p\rightarrow\|u+v\|_p$$

Is true that $$\|v_n\|_p\rightarrow\|v\|_p$$

Thanks

Are you sure this is what you want to ask? Convergence of norm and convergence in norm are different things... — Miha Habič, Nov 10 '12 at 20:43

score 2 · Accepted Answer · answered Nov 10 '12 at 21:10

2

This is false. Consider the constant functions $u=-1,v_n=1+(-1)^n,v=0$.

answered Nov 10 '12 at 21:10

Miha Habič

1 Answers1