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Let S be a subset of a Hilbert space H. Prove that Span(S) is dense in H if and only ifS⊥ = 0.

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Let S be a subset of a Hilbert space H. Prove that Span(S) is dense in H if and only ifS⊥ = 0.
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Solution 1

Sure, here is the proof:

First, let's define some terms:

  • Span(S) is the smallest subspace of H that contains S.
  • S⊥ is the orthogonal complement of S, which is the set of all vectors in H that are orthogonal to every vector in S.
  • A subset of a topological space is said to be dense if its clos Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI  is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI  is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI  is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI  is a powerful AI-powered study tool designed to help you to solve study problem.
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