Interchanging Linear Operator and Infinite Sum

[logarithmic]

Suppose that $x\in H$, where H is a Hilbert space. Then x has an orthogonal decomposition $x = \sum_{i=0}^\infty x_i$.

I have a linear operator P (more specifically a projection operator), and I want to write:
$P(x) = \sum_{i=0}^\infty P(x_i)$.

How can I justify taking the operator inside the infinite sum?

 

[Erland]

This is true if the linear operator P is bounded. Otherwise, it might be false.