Will the resulting Euclidean space be Hilbertian? In the linear space of sequences $x=( x_{1}, x_{2}, ... ), (x_{k}\in \mathbb{R})$ such that $\sum_{k=1}^{\infty }x_{k}^{2}<\infty$. 
Let $(x,y)=\sum_{k=1}^{\infty }\lambda_{k}x_{k}y_{k}$, where $\lambda_{k}\in \mathbb{R}, 0<\lambda_{k}<1$. Will the resulting Euclidean space be Hilbertian?

The form $(x,y)$ is bilinear symmetric i.e. a scalar product in 
$$
H=\{( x_{1}, x_{2}, ... )\in\mathbb R^\infty\;|\,\sum_{k=1}^{\infty }x_{k}^{2}<\infty\}
$$
Whether $H$ is Hilbertian or not depends on $\lambda_k$. For eample, if
$$
\lambda_k=2^{-k},
$$
then $H$ is not hilbertian. In fact, the sequence 
$$
x^{(k)}=(1,\ldots,1,0\ldots)\qquad (\textrm {with k one's})
$$
is Cauchy in the norm of the $\lambda$-scalar product, but has no limit in $H$.
Instead
$$
H'=\{( x_{1}, x_{2}, ... )\in\mathbb R^\infty\;|\,\sum_{k=1}^{\infty }\lambda_kx_{k}^{2}<\infty\}
$$
is Hilbertian
