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Verify that a Hilbert space orthonormal basis in a finite dimensional Hilbert space is the same as an orthonormal basis in the sense of linear algebra.

Here is what I know. Hilbert space orthonormal basis is defined to be an orthonormal set such that the closure of its span is dense. But in finite dimensional case the span is closed already.

What exactly do I need to show now. This question seems quite straightforword but I do not know what to do next. Could anyone help me, please? Thank you!

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  • $\begingroup$ The only way for a set to be closed and dense is if it's everything. $\endgroup$ – Adam Hughes Nov 11 '14 at 3:29
  • $\begingroup$ @AdamHughes What is the difference between the Hilbert space orthonormal set and the linear algebra orthonormal set? $\endgroup$ – LaTeXFan Nov 11 '14 at 3:31
  • $\begingroup$ The difference between a Hilert space orthonormal basis and a classical linear algebra one is that the sums are infinite and require limits to define. When the sums are all finite because the dimension is finite, then everything reduces to the classical case of linear algebra without norm convergence requirements. $\endgroup$ – DisintegratingByParts Nov 12 '14 at 11:10

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