How can one see that $I$ is an infinite projection in $B(\mathcal{H})$, where $\mathcal{H}$ is an inseparable Hilbert space?



On projection theory for inseparable Hilbert spaces
Let $\beta$ be a Hilbert basis for $H$. If $\tilde\beta$ is a finite subset of $\beta$ and $f:\beta\rightarrow \beta\setminus\tilde\beta$ is a bijection, then 
$$U:H\rightarrow H,\quad v\mapsto f(v)$$
introduce a partial isometry in $B(H)$ whose initial projection is $I$ and its final projection is $I-P$, where $P$ is a projection on the closed subspace generated by $\tilde\beta$. So $I\sim I-P\neq I$ and that mean $I$ is infinite. 


Also, if we choose disjoint subsets $\beta\_1,\beta\_2$ such that $\beta\_1\cup\beta\_2=\beta$ and $\beta$, $\beta\_1$ and $\beta\_2$ have same cardinals, the argument like the one above shows that $I$ is properly infinite projection. 


