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I asked this question in a comment when I realised that the answerer is away until September so I am posting it here in a new thread.

I've been thinking about the Sobolev embedding theorem, given as follows: If $k > l + d/2$ then we can continuously extend the inclusion $C^\infty (\mathbb T^d) \hookrightarrow C^l (\mathbb T^d)$ to $H^k (\mathbb T^d) \hookrightarrow C^l(\mathbb T^d) $ where $\mathbb T^d$ is the $d$-dimensional torus and $H^k$ is the closure of $C^\infty$ with respect to the norm $\|(D^\alpha f)_\alpha \| = \sqrt{ \sum_\alpha \|D^\alpha f\|^2} $.

Can you tell me if this is correct?

(i) By definition of $H^k$ we can uniquely and continuously extend any continuous linear operator $T$ that has domain $C^\infty (\mathbb T^d)$ to all of $H^k$.

(ii) What the Sobolev embedding theorem gives us is a continuous inclusion $i: H^k \hookrightarrow C^l$ so that given a continuous linear operator $T: C^l \to X$ (to any linear normed space $X$) we can apply $T$ to $H^k$ via $T \circ i$.

I think I used to mix up (i) and (ii) and now I think that these are two different facts, independent of each other. Thanks for your help.

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I'm assuming you are thinking of $C^\infty$ as a normed space (with the Sobolev norm) in i). Then i) is correct yes, but has nothing to do with Sobolev embedding.

The Sobolev embedding theorem(s) is/are a statement about the regularity of the functions in Sobolev spaces, they claim that there is a smooth/continuous function representing an element of a space obtained by completing a space of smooth functions using integral norms. This is important when you want to prove smoothness of solutions of differential equations, the existence of which is often rather easy to prove in Sobolove spaces. (The statement of the embedding theorem usually includes a statement about the compactness of the embedding (when applicable), which is also useful for proving existence theorems).

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    What I wrote above is non-sense. We have $f \mapsto (D^\alpha f)_\alpha$ and then we take the closure of this set.2012-08-20