There is a definition of Hilbert basis in this paper (chapter 1.2) that says : $(e_{i})$ is a Hilbert basis iff $$ (1). (e_{i}\dagger e_{j})=\delta_{i,j}$$ $$(2). \Sigma e_{i}e_{i}\dagger = I$$ where $I$ represents the identity matrix.
My question here is: isn't the (2). requierement enough for $(e_{i})$ to be an Hilbert basis? In other terms: does (2). imply (1). ?
In finite dimension I think it is the case. Because if we put the $(e_{i})$ vectors as columns in a matrix $H$ we have: $$\Sigma e_{i}e_{i}\dagger = I \iff HH\dagger = I $$ which means $ H\dagger H= I $ which I think implies $ (e_{i}\dagger e_{j})=\delta_{i,j}$
I may be wrong... Thanks in advance