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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 $$\|v_n\|_p\rightarrow\|v\|_p$$


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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
Yes it is @MihaHabič – Tomás Nov 10 '12 at 20:47
up vote 2 down vote accepted

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

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