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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 $(x_{n}-x,\ y)$ $\rightarrow$ $0$ whenever $y$ in $E$.

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

and

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

(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 $(E,(.,.))$ .)

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 $(E,(.,.))$ .)

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 $(x_{n}-x\, y$) $\rightarrow$ $0$ whenever $y$ in $E$.

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

and

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

(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 $(E,(.,.))$ .)

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 $(x_{n}-x,\ y)$ $\rightarrow$ $0$ whenever $y$ in $E$.

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

and

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

(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 $(E,(.,.))$ .)