Let $H_n$ be Hilbert spaces for $n\in \mathbb{N}$. Then define
$H:=\bigoplus_{n=1}^\infty H_n$
with inner product
$\langle h,\tilde{h}\rangle_H$ $=\sum_{n\in\mathbb{N}}\langle h_n,\tilde{h_n}\rangle_{H_n}$
Then $H$ is again a Hilbert space. Does it satisfy the universal property of the coproduct in the category of Hilbert spaces and maps that are bounded by one. I think the map that we get from the universal property should be of the form
$q(h):=\sum_{n\in\mathbb{N}} f_n(h_n)$
for maps $f_n:H_n\to T$ with $||f_n||\leq 1$. My problem is to show that this map is well defined, that is $||q(h)||\leq||h||$. Using the usual estimates for $f_n$, Cauchy schwartz and linearity of the inner product I can only get
$||q(h)||^2\leq \sum_{n\in\mathbb{N}}\langle h_n,h_n\rangle^\frac{1}{2}$
The left hand side is not the norm induced by the inner product I defined above. Can somebody help? Or does it not satisfy the universal property of the coproduct?