What is the difference between a Hilbert space and Euclidean space? According to Wikipedia, 

Hilbert space [...] extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions

However, the article on Euclidean space states already refers to 

the n-dimensional Euclidean space.

This would imply that Hilbert space and Euclidean space are synonymous, which seems silly.
What exactly is the difference between Hilbert space and Euclidean space? What would be an example of a non-Euclidean Hilbert space?

Hilbert space: a vector space together with an inner product, which is a Banach space with respect to the norm induced by the inner product 
Euclidean space: a subset of $\mathbb R^n$ for some whole number $n$
A non-euclidean Hilbert space: $\ell_2(\mathbb R)$, the space of square summable real sequences, with the inner product $((x_n),(y_n)) = \sum_{n=1}^{\infty}x_n y_n$
