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On the space $C^\infty(S^1,\mathbb R)$, for each $n\in \mathbb N$, define $$p_N(\gamma)= \max\{|f^{(k)}(t): t\in S^1, k\leq N\}$$

Topology of all norms above define a metrizable locally convex topology (in fact Frechet space) on this space [Rudin Functional analysis page 35].

How to calculate dual space to this space,

For dual space, I mean set of all continuous linear functional on $C^\infty(S^1, M)$ with norms $$p'_M(f)= \sup_{\gamma\in M\subset C^\infty(S^1,\mathbb R)}|f(\gamma)|$$ and $M$ runs through all bounded subsets of $L$.

My background and others: I do not have enough practice and knowledge of functional analysis course.. Hence i will be happy if i get reference reading for this so that i can calculate dual myself.

What are the books/topic name which i should read to get comfortable in calculating these type questions

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I get the impression that the completion of this space is $H^N$. If this is true, the dual is the dual of $H^N$, which is $H^N$ or $H^{-N}$ depending on how you identify it (if I remember right). –  user45150 Jan 31 '13 at 7:09
    
Also how did this question come up? I think that would give at least some direction responders can proceed in. –  user45150 Jan 31 '13 at 7:12
    
@user45150 thanks for the comment, but space is complete.. I edited the question, i guess you confused that i am fixing $N$.. but topology is coming from all these countable seminorms. –  Junu Jan 31 '13 at 7:50
    
This is the space of all distributions on the circle if I'm correct. –  Ahriman Jan 31 '13 at 7:56
    
@Ahriman, How did you get that.. can you please explain or give some reference.. –  Junu Jan 31 '13 at 7:57

1 Answer 1

For a compact space, this is indeed the space of all distributions, which has the topology of pointwise convergence. To check that something is a continuous functional, you can check continuity with respect to the seminorms in $C^{\infty}(M)$ separately (this construction can be generalised to smooth sections of vector bundles where you have connections).

The second norm that you write gives the strong topology or the topology of uniform convergence on bounded subsets. In general they are not the same, though they coincide for Banach spaces. When $M$ is not compact, the topology on $C_c(M)$ is more complicated (for example, see Richard Melrose's notes on Differential Analysis pp. 40). A general good reference is Michael E. Taylor's Partial Differential Equations, Vol I, appendix A, which has an excellent condensed account of the functional analytic tools that are normally used.

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