Bounded Linear Operator on Hilbert Space Direct Sum

Theorem
Let $\left({H_n}\right)_{n \in \N}$ be a sequence of Hilbert spaces.

Denote by $H = \displaystyle \bigoplus_{n \mathop = 1}^\infty H_n$ their Hilbert space direct sum.

For each $n \in \N$, let $T_n \in B \left({H_n}\right)$ be a bounded linear operator.

Suppose that one has $\displaystyle \sup_{n \mathop \in \N} \, \left\Vert{T_n}\right\Vert < \infty$, where $\left\Vert{\cdot}\right\Vert$ signifies the operator norm.

Define $T: H \to H$ by:


 * $\forall h = \left({h_n}\right)_{n \in \N}: T h = \left({T_n h_n}\right)_{n \in \N} \in H$

Then $T \in B \left({H}\right)$ is a bounded linear operator.

Proof
Let $c = \displaystyle \sup_{n \mathop \in \N} \, \left\Vert{T_n}\right\Vert$.

By assumption, $c < \infty$.

Let $h = \left({h_n}\right)_{n \in \N} \in H$ be arbitrary.

Then:

In summary:


 * $\left\Vert{ T h }\right\Vert_H^2 \le c^2 \left\Vert{ h }\right\Vert_H^2$

It follows that $T$ is bounded.