Given X and another Hilbert space Y , show that the product space X×Y is also a Hilbert space, with inner product ⟨(x, y), (u, v)⟩ = ⟨x, u⟩ + ⟨y, v⟩.

Question

I am trying to prove the above question. I know that I need to some how show that it satisfies the following axioms

(i) $\langle x,x\rangle \geq 0$ and $\langle x,x\rangle=0$ if and only if $x=0$,

(ii) $\langle x,y\rangle = \langle y,x\rangle$,

(iii) $\langle \lambda x,y\rangle = \lambda \langle x,y\rangle$,

(iv) $\langle x,y+z\rangle =\langle x,y\rangle + \langle x,z\rangle$

but I just know now how to relate it since the inner product in the question has two terms for example $\langle (x,y),(u,v)\rangle$ instead of just $\langle x,y\rangle$.

I am hoping someone can just point me in the right direct with a hint or some direction.

Thanks you

Answer 1

You can check the axioms directly. For example, you have, for $x\in X$ and $y\in Y$,

i) $\langle (x,y),(x,y)\rangle = \langle x,x\rangle + \langle y,y\rangle \geq 0$.