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I was just working on some problems regarding the mathematical formalism of QM, and while trying to finish a proof, I realized that I am not sure if the following fact is always true:

Suppose that we have two linear operators A and B acting over some vector space. Consider a state ket | [itex]\psi[/itex] >

I am wondering if

< [itex]\psi[/itex] | (A+B) | [itex]\psi[/itex] > = < [itex]\psi[/itex] | A | [itex]\psi[/itex] > + < [itex]\psi[/itex] | B | [itex]\psi[/itex] >

is always true?

I am thinking that it IS true.

My attempt at the problem, is of course to try and show that

(A+B) | [itex]\psi[/itex] > = A | [itex]\psi[/itex] > + B | [itex]\psi[/itex] >

But I am having trouble finding a definition which will confirm this to always be true.

I feel like I am completely overlooking something. Does anyone have a helpful hint for me? ANy literature to point me to? My linear algebra books are failing me on this one, at first glance.