Today I am working with Functional Analysis regarding Hilbert Spaces from the notes. Let $H_1$ and $H_2$ be Hilbert Spaces. The direct sum $H_1\oplus H_2$ is the vector spaces $H_1\times H_2$ with the following inner product,
$$\langle (x,y)|(x',y')\rangle:=\langle x|x' \rangle_{H_1}+\langle y|y' \rangle_{H_2} ,$$ where $(x,y),(x',y')\in H_1\times H_2 $.
Well I know the definition of an inner product, i.e. linearity, symmetry, and positive definiteness however I cannot see how to do it with this direct sum. Anyone who can give me a hint or maybe show one of them and then I can try to work the rest out. Thanks in advance.
Notice this is NOT homework. It is just me that wants to convince myself that this is a inner product.
Edit: I've shown that this indeed defines an inner product. How about norm? Is it true that the norm of $H_1\oplus H_2$ is $ \|\langle x,x'\rangle,\langle y,y'\rangle\|=(\|\langle x,x' \rangle\|^2+\|\langle y,y' \rangle\|^2)^\frac{1}{2}$