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# Hilbert Space and Subspace

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Problem. Show that if is an orthonormal set in a Hilbert space H, then the set of all vectors of the form is a subspace of H.

Hint: Take a Cauchy sequence , where . Set and show that is a Cauchy sequence in .

Please see the attached file for full problem description.

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A Hilbert Space and Subspace are investigated. The solution is detailed and well presented.

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Problem. Show that if is an orthonormal set in a Hilbert space H, then the set of all vectors of the form is a subspace of H.

Hint: Take a Cauchy sequence , where . Set and show that is a Cauchy sequence in .

Solution:

I will show you a short and very easy solution based on the properties of a Hilbert space (that is normed, complete and defined by a dot product)
A "complete" space means that every Cauchy sequence of the ...

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