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## Main Question or Discussion Point

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

I have a linear operator P (more specifically a projection operator), and I want to write:

[itex]P(x) = \sum_{i=0}^\infty P(x_i)[/itex].

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

I have a linear operator P (more specifically a projection operator), and I want to write:

[itex]P(x) = \sum_{i=0}^\infty P(x_i)[/itex].

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