Continuous linear functionals on $\mathbb{R}^\mathbb{N}$

Question

What are the continuous linear functionals on $\mathbb{R}^\mathbb{N}$, when equipped with the product topology? In particular, do they depend on only finitely many coordinates? If not, for what (natural) topology would that be true?

Answer 1

Let $f$ be a continuous linear functional on $\mathbb R^{\mathbb N}$. Restricted to $l^{1}$ we get a continuous linear functional so there exists $\{a_n\} \in l^{\infty}$ such that $f(x_n)=\sum x_n a_n$ whenever $\sum |x_i| < \infty$. Now $f(x_1,x_2,...)=\lim f(x_1,x_2,...,x_n,0,0...)=\lim \sum _{i=1}^{n} a_ix_i$. Hence the series $\sum a_i x_i$ converges for all $(x_i) \in \mathbb R^{\mathbb N}$. From this it is trivial to see that $a_n=0$ for $n$ sufficiently large.