Hello,

i try to understand a few isomorphisms, which i have found without comment in a text:

Let H be a Hilbert space and

Then we have the following identities:

I know that we can decompose the Space H by

And by the isomorphism above (1) we get

Analogue we have also a decomposition

But then the text says:

Are these equations always true? I mean T is bounded in this case, but isn't it necessary that T is injective or something like that? It would be nice, if someone can explain those eqations to me.

Regards