Let $l^2$ be the set of sequences $x = (x_n)_{n\in\mathbb{N}}$ ($x_n \in \mathbb{C}$) such that $\sum_{k\in\mathbb{N}} \left|x_k\right|^2 < \infty$, how can I prove that $l^2$ is a Hilbert space (with dot-product $\left(x,y\right) = \sum_{k\in\mathbb{N}} x_k\overline{y_k}$). This is a standard textbook exercise: apparently this is easy and, even to me, it seems self-evident. However, I don't know what to do with the infinite sum.
How to prove that square-summable sequences form a Hilbert space?
9
$\begingroup$
sequences-and-series
hilbert-spaces
-
5"Hilbert space" encompasses a number of properties. Which one is causing you trouble? – 2011-03-09
-
0Well, for finite sums, I have no problems; but how do I know, for example, if $\overline{\sum_{k\in\mathbb{N}} a_k} = \sum_{k\in\mathbb{N}} \overline{a_k}$ (for showing that $\overline{(y,x)} = (x,y)$)? Moreover, I have trouble showing that every Cauchy sequences is convergent in $l^2$. – 2011-03-09
-
0Use the fact that, by definition, $\sum_{k\in\mathbb{N}a_k} = \lim\limits_{n\to\infty}\sum_{k=1}^n a_k$. Work with the finite partial sums, and use the properties of limits to show the properties you want also holds for the series. – 2011-03-09
-
0Great, that solves half my problem! – 2011-03-09
-
1LaTeX tip "of the day": try using `\ell_p` for the sequence space $\ell_p$ – 2011-03-09