Do Hilbert Space Isomorphism Map Dense Sets to Dense Sets?

  • #1
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Main Question or Discussion Point

Suppose that H, K are Hilbert spaces, and A : H -> K is a bounded linear operator and an isomorphism.

If X is a dense set in H, then is A(X) a dense set in K?

Any references to texts would also be helpful.
 

Answers and Replies

  • #2
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It is true. Suppose, that A(X) is not dense. Then let V be a non-empty open set in K \ A(X). The pullback of V by A is open (A is bounded) and not empty (A is isomorphism), and by definition it is not in X, which contradicts the fact, that X is dense in H.
 
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