Question

let H be finite separable hilbert space. show that H is unitarily isomorphic to cʳ for some r ∊ N.

Answer 1

H be finite separable hilbert space

 U : H → H on a Hilbert space H that satisfies U*U = UU* = I, where U* is the adjoint of U, and I : H → H is the identity operator.

 A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold:

U is surjective

. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold:

the range of U is dense in H,

H is unitarily isomorphic to cʳ for some r ∊ N.

$a=(a_i) \space i\epsilon n \space {\epsilon} f(n) \space and \space Cx =a , \space \sum _{i=a}^{\infin}a_ie_i$

Thus every separable Hilbert space can be mapped onto f(n) by a unitary operator. We say that every separable Hilbert space is unitarily isomorphic to f(n)

f(n)=c^r and r$\epsilon$ N