Sobolev embedding

Question

In an exercise I am asked to prove the following statement: The embedding $$T:W^{1,1}(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n), u\mapsto u$$ is continuous. Using the Gagliardo-Nierenberg inequality $\|u\|_{W^{1,1}(\mathbb{R}^n)}\geq\|u\|_{L^2(\mathbb{R}^n)}$, $u\in\mathcal{C}^1_c(\mathbb{R}^n)$, I managed to prove this result for the domain $W^{1,1}_0(\mathbb{R}^n)$ where I can approximate by test functions. Does someone know how to approach the general case? Why is the embedding well-defined?

Answer 1

I am not sure you need this answer or not. But I think @PhoemueX already give you one.

In general, keep in mind that $W^{1,p}=W_0^{1,p}$. (prove it! good exercise). Then the rest is clear.