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.