Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$ on which $a$ is negative-definite.

If $V$ is infinite-dimensional, it can be very hard to check every negative-definite subspace and compare their dimensions. Much easier is to exhibit a subspace $V_- \subseteq V$ on which $a$ is negative-definite, which is not contained within any other negative subspace. But how do I know that the dimension of this negative subspace is the maximal dimension of any negative subspace? The approach I thought worked fails; see this question.

Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$ on which $a$ is negative-definite.

If $V$ is infinite-dimensional, it can be very hard to check every negative-definite subspace and compare their dimensions. Much easier is to exhibit a subspace $V_- \subseteq V$ on which $a$ is negative-definite, which is not contained within any other negative subspace. But how do I know that the dimension of this negative subspace is the maximal dimension of any negative subspace? The approach I thought worked fails; see this question.