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Let H be a Hilbert space. Let F be a subset of H.

F is dense in H if:

<f,h>=0 for all f in F => h=0

Now take an orthonormal set (e

[itex]\bigcup[/itex]span(e

=>

g=Ʃ<g,e

Now first of all:

Why do we need the union of the spans in the above? What is the difference between the span of the sequence of countably many units vectors and the union of the spans?

And then: Why is the above implication obvious?

F is dense in H if:

<f,h>=0 for all f in F => h=0

Now take an orthonormal set (e

_{k}) (this is a countable sequence indexed by k) in H. My book says that obviously:[itex]\bigcup[/itex]span(e

_{k}) is dense in H (the union runs over all k)=>

g=Ʃ<g,e

_{k}>e_{k}Now first of all:

Why do we need the union of the spans in the above? What is the difference between the span of the sequence of countably many units vectors and the union of the spans?

And then: Why is the above implication obvious?

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