What's so great about continuous embeddings?

I often see something like "$V \subset H$ is continuous embedding", meaning that the inclusion map $i:V \to H$ defined $i(v) = v \in H$ for $v \in V$ is continuous: so $$|i(v)|_H \leq C|v|_V$$ holds for some constant $C$.

Why is this so important?

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1 Answer

I assume you are talking about Banach spaces here.

Continuous embeddings are important because they preserve the structure of the smaller Banach space and allow us to regard it as a subspace of the larger.

Convergence of the sequence $(x_n)$ in the space $V$ implies convergence of $(i(x_n))$ in the space $i(V)$ for instance.

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