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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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1 Answer 1

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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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Thank you, Thomas! Yes, I am thinking of $C^\infty$ as sequences in $\prod_{N(k)} L^2$ of the form $(f, 0, 0 \dots)$ where $f \in C^\infty$ with the Sobolev norm I gave in the theorem. Did you see Davide's comment in the thread I linked and my question in response to it? I'm asking because if (i) is correct, his comment seems incorrect. Sorry, I forgot to include this in the question. –  Rudy the Reindeer Aug 18 '12 at 12:42
    
Perhaps I should say vectors not sequences since $N(k) < \infty$. –  Rudy the Reindeer Aug 18 '12 at 12:47
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@Matt: No, I guess Davide is correct, if I got the intent of your question correctly. If you have a property which is true in $C^\infty$ and you want to have that property in the Sobolev space, you have to prove it (most naturally using approximation in the norm with respect to which you took the closure, showing that the property is preseved when taking limits in that norm). Take, as a simple example that, for any $k$, the $k-th$ derivative of a $C^\infty$ function exists and is Lipshitz (on a compact domain). This fails to be true rather soon for the Sobolev space if $k$ increases. –  user20266 Aug 18 '12 at 13:02
    
@Matt: another remark regarding your comment to Davide's answer: even though you are seemingly dealing mostly whith linear maps and structures right now, not everything you can do to a smooth function is expressible by means of linear operators. Maybe this is why you got deluded... –  user20266 Aug 18 '12 at 13:17
    
Thank you very much! –  Rudy the Reindeer Aug 18 '12 at 20:20

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