Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be two Hilbert spaces. We define the space $\mathcal{H}$ by
$$\mathcal{H} = \mathcal{H}_1 \oplus \mathcal{H}_2= \{ x=(x_1,x_2) : x_1 \in \mathcal{H}_1
,x_2 \in \mathcal{H}_2 \} $$
and
$$\forall x,y \in \mathcal{H}, x=(x_1,x_2),y=(y_1,y_2), <x,y>_{\mathcal{H}} = <x_1,y_1>_{\mathcal{H}_1}
+<x_2,y_2>\mathcal{H}_2 $$
Prove that $\mathcal{H}$ is a hilbert space
Need to show that it is a 1) an inner product space rules are obeyed and 2) that it is complete. Having issues with showing that it s complete. So if it is cauchy it converges in the space. Having trouble to notate that it is cauchy with $<.,.>$
should used diff notation that $\oplus$
Attempt 1
Suppose that $(X_n,Y_n) \in H$ are Cauchy so $X_n=(x_{n1},x_{n2})$ and $Y_n=(y_{n1},y_{n2})$. So $$<X_n,Y_n>_\mathcal{H}=<x_{n1},y_{n1}>_{\mathcal{H_1}}+ <x_{n2},y_{n2}>_{\mathcal{H_2}} $$ So each $<x_{n1},y_{n1}>_{\mathcal{H_1}}$ are cauchy then they converge to some $<x_1,y_1>_\mathcal{H_1}$ and $<x_2,y_2>_\mathcal{H_1}$ that is $<X,Y> \in \mathcal{H}$
I am not sure about my notation with Cauchy. I dont feel confortable with the def of cauchy in Inner product spaces that is my real question
The norm of $\mathcal{H}_1 \oplus \mathcal{H}_2$ is the sum of the norms of the components.
So if $(x_n,y_n)\in \mathcal{H}_1 \oplus \mathcal{H}_2$ is a Cauchy sequence, both $x_n$ and $y_n$ are Cauchy in their spaces, so they are convergent. From where it follows that $(x_n,y_n)$ is convergent.