Let $E$ be a real linear space, endowed with a non-degenerate symmetric bilinear form $(.,.)$.

Suppose that the [indefinite] inner product space $(E,(.,.))$ satisfies the following [sequential] properties:

(**WSC**) If the sequence { ${(x_{n},\ y)}$ } is Cauchy for each $y$ in $E$, then there
exists some $x$ in $E$ such that $\left(x_{n}-x,y\right)$ $\rightarrow$ $0$
whenever $y$ in $E$.

(That is to say, $(E,(.,.))$ is *weakly sequentially complete*.)

and

(**DPG**) If $\left(x_{n},y\right)$ $\rightarrow$ $0$ for every $y$ in $E$,
then $\left(x_{n},x_{n}\right)$ $\rightarrow$ $0$.

(That would be sort of "*Dunford-Pettis & Grothendieck*'' property for indefinite inner product spaces.)

Suppose also that $|E|$ is "big enough'' (at least $|E|$ $>|\mathbb{R}|$).

**Conjecture**. $(E,(.,.))$ contains an infinite-dimensional Hilbert
subspace.

(That is, there exists a linear isometry from $(\ell^{2},<.,.>)$ into $(E,(.,.))$ .)