Sequences in $H^1$ which are orthonormal w.r.t. $L^2$ product

Question

I'm currently reading about Jacobi fields and conjugate points in Riemannian Geometry and Global Analysis by Jost, and a few details are losing me (probably because I have a weak background in functional analysis).

Let $c: [a, b] \to M$ be a geodesic on a Riemannian manifold $(M, \left< \cdot, \cdot \right>)$. Define $\mathcal{V}_c := \Gamma(c^* TM)$, the space of vector fields along $c$, and $\hat{\mathcal{V}}_c$ the vector fields $V \in \hat{\mathcal{V}}_c$ satisfying $V(a) = V(b) = 0$. The index form is defined as the symmetric bilinear form $I$ such that

$I(X, Y) = \int_a^b \left[\left<D_t X, D_t Y\right> - \left<R(\dot{\gamma}, X)Y, \dot{\gamma}\right>\right]dt$ where $X, Y \in \mathcal{V}_c$.

Jost then defines a norm on $\hat{\mathcal{V}}_c$ by $||X||^2 = \int_a^b \left[\left<D_t X, D_t X\right> + \left<X, X\right> \right]dt$, and lets $\hat{H}_c^1$ be the completion of $\hat{\mathcal{V}}_c$ w.r.t. $||\cdot||$. This space is easily identified with the Sobolev space $H^1_0(I, \mathbb{R}^n)$. He then reinterprets the index form $I$ as a symmetric bilinear form on $\hat{H}_c^1$.

The index of $c$, Ind$(c)$, is defined as the dimension of the largest subspace of $\hat{H}_c^1$ on which $I$ is negative definite. He shows that Ind$(c)$ is finite, but there's a core detail I don't understand. Namely, that if the Ind$(c)$ were not finite, then there exists a sequence $(X_n)\subset \hat{H}_c^1$ such that $I(X_n, X_n) \le 0$ and $(X_n)$ is orthonormal w.r.t. the $L^2$ product.

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The fact that $(X_n)$ is orthonormal w.r.t. the $L^2$ product is hugely important in the proof. I suppose that he put the norm on $\hat{\mathcal{V}}_c$ just so he could eventually define orthonormality. So my questions are:

1. Why can we find such an orthonormal sequence?
2. Is there anything "lost" in this construction? It feels a bit arbitrary to redefine $I$ on $\hat{H}_c^1$. For example, we could also define the index as the largest subspace of $\hat{\mathcal{V}}_c$ on which $I$ is negative-definite. Are these two notions of index necessarily equivalent?