Interchanging Linear Operator and Infinite Sum

  • #1
logarithmic
107
0
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?
 

Answers and Replies

  • #2
Erland
Science Advisor
741
138
This is true if the linear operator P is bounded. Otherwise, it might be false.
 

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