The difference between closed linear span and linear span in Hilbert spaces.

Question

$H$ is a Hilbert space and $M$ is an orthonormal set(not necessarily finite). What is the definition of:
1)closed linear span of $M$
2)the linear span of $M$
3)the closure of a linear span

I was reading Stein's Real Analysis .He used the concept span in the proof of Spectral Theorem in Page 193. But I can not find the definition of span(infinite cases) in this book.

Answer 1

Closed linear span is equal (by definition) to closure of linear span