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
Best Answer
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.