Is this space Hilbert space?

Question

Let's say we have linear space $$ \ell^2 := \{x=(x_1, x_2, ..): \sum_{i=1}^{\infty} x_i^2 < \infty \} $$ with $$ <x,y> := \sum_{i=i}^{\infty} \lambda_i x_i y_i $$ where $\lambda_i \in \mathbb{R}$, $ 0 < \lambda_i <1$.

Is this space Hilbert space and is this $<x,y>$ Euclidean?

Answer 1

The function $\langle x, y \rangle = \sum_i \lambda_i x_iy_i$ does define and inner-product; I assume that this is what you mean by "Euclidean".

That being said, the space $\ell^2$ might not be a Hilbert space under this inner product since it might fail to be complete. For instance, if we take $\lambda_i = 2^{-i}$ and define the sequence $(x^{(k)})_{k \in \Bbb N}$ by $$ x^{(k)}(j) = \begin{cases} 1 & j \leq k\\ 0 & j > k, \end{cases} $$ then we find that $x^{(k)}$ is Cauchy relative to the norm induced by this new inner product, but fails to have a limit in $\ell^2$.