subspaces $U,V\in G(H)$ can also be obtained as follows. Let $S_U$ denote the unit sphere in $U$ and $d(u,V)=\inf_{v\in V}{\|u-v\|}$. Then for $\delta(U,V)=\sup_{u\in S_U} d(u,V)$,

$$\begin{aligned}
\label{gap}
d_G(U,V)=\max\{\delta(U,V),\delta(V,U)\},\end{aligned}$$

which explains why $d_G(U,V)$ is called the *gap* between $U$ and $V$.\
We now consider operators $T:\mathcal{D}(T)\subset H\rightarrow H$ which we assume to be defined on a dense domain $\mathcal{D}(T)$. Let us recall that $T$ is called *closed* if its graph $\operatorname{graph}(T)$ is closed in $H\times H$. If we denote by $\mathcal{C}(H)$ the set of all closed operators, then the gap-metric on $H\times H$ induces a metric on $\mathcal{C}(H)$ by

$$d_G(T,S)=d_G(\operatorname{graph}(T),\operatorname{graph}(S)), \quad S,T\in\mathcal{C}(H).$$ As the adjoint of a densely defined operator is closed, every selfadjoint operator on $H$ belongs to the metric space $\mathcal{C}(H)$. Moreover, let us recall that a closed operator $T$ is called *Fredholm* if its kernel and cokernel are of finite dimension. In what follows, we denote the subset of $\mathcal{C}(H)$ consisting of all $T$ which are selfadjoint and Fredholm by $\mathcal{CF}^\textup{sa}(H)$. It is well known that the spectrum $\sigma(T)$ of every selfadjoint operator is real. Moreover, if $T\in\mathcal{CF}^\textup{sa}(H)$ then $0$ is either in the resolvent set or an isolated eigenvalue of finite multiplicity (see e.g. [@Fredholm Lemma 2.2.5]).\
