Find the vector in $\ell^2(\mathbb N)$ corresponding to $f$ as in the Riesz Representation Theorem

Question

I've been trying to get my head around the Riesz Representation Theorem and I'm stuck on this question.

Let $N \in \mathbb N$ and define $f: \ell^2(\mathbb N) \to \mathbb R$ by $f((a_n)_{n=1}^{\infty}=a_N $.

1) Prove that $f$ is continuous linear functional on $\ell^2(\mathbb N)$ and 2) find the vector in $\ell^2(\mathbb N)$ corresponding to $f$ as in the Riesz Representation Theorem.

I can do the first part 1) but part 2) is where I get stuck. I've looked up various proofs that involve integration but they don't really make sense to me.

What would be the easiest way to figure this out?

Any help would be great!

Answer 1

As $|f((a_n))| = |a_N| = \sqrt{a_N^2} \leq \sqrt{\sum_{n= 1}^\infty a_n^2}= \|(a_n)\|_{l^2}$, $f$ is continuos. Note now that $$f((a_n)) = \sum_{n= 1}^\infty a_n b_n = \langle(a_n),(b_n)\rangle_{l^2},$$ where $b_n =0$ if $n\neq N$ and $b_N = 1$.