Embeddings and Imbeddings

Question

I am studying Sobolev Spaces and I am very confused about the following:

What are the difference between the Embeddings and Imbeddings of spaces?

If I have that: $W^{k,p}\to W^{m,q} $ is a compact imbedding, so it means that $W^{k,p}\subset W^{m,q} $ or not ?

Answer 1

AFAIK Gae. S. in comments is correct: embeddings and imbeddings are alternate spellings of each other.

A compact im/embedding $\iota : X\to Y$ between Banach spaces, say (which already means $\iota (X) \subset Y$) is a map satisfying the following:

• (embedding property) $\iota$ is injective,
• (Continuity) we have the norm bound $ \|\iota(u)\|_Y \le C \|u\|_X$ for some $C>0$,
• (Compactness) if $u_n$ is a bounded sequence of $X$, then $\iota(u_n)$ has a convergent subsequence in $Y$.

So what you said is true: $\iota(W^{k,p})\subset W^{m,q}$, but you have even a quantitative control on norms and the ability to extract convergent subsequences in the 'larger' space. Note that it is common practice to not give a symbol to the embedding map, and say that $X$ is a subset of $Y$ (this is the identification of $X$ with its embedding in $Y$).