MHB How Does Sobolev Space Boundedness Relate to Different Norms in $R^n$?

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If $ r<s<t $ then for any $ ϵ>0 $there exists $ C>0 $ such that $ ∥f∥_{(s)}≤ϵ∥f∥_{(t)}+C∥f∥_{(r)} $for all $f∈H_t $

Can you please tell me how to start thinking of this problem? I really feel stuck and don't know where to start!
 
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Welcome, Danny! (Wave)

Some information is missing in the problem statement. What is the domain of $f$?
 
$f$ is a tempered distribution
 
What I mean is this: $f\in H^t(X)$ for some space $X$ -- what is $X$? It makes a difference, since, e.g., integration on the torus is different from integration on $\Bbb R^n$.
 
$R^n$
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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