Is the sequence space over natural numbers $\ell^2(\mathbb{N})$ a Hilbert space?

Question

In my homework for functional analysis, we had $\mathcal{H} = \ell^2(\mathbb{N})$. But the usual sequence space is defined to be a vector space over complex or real numbers. I'm not sure how $\ell^2(\mathbb{N})$ is a Hilbert space.

Answer 1

$\ell^2(\mathbb{N})$ is the space of the (complex or real) sequences $x = (x_n)_{n\in\Bbb N}$ s.t. $$\sum_{n\in\Bbb N}|x_n|^2 < \infty.$$ The scalar product is $$(x,y) = \sum_{n\in\Bbb N}x_n\bar{y_n}.$$