Page 369 - Advanced Linear Algebra
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Hilbert Spaces  353



            13.  Prove that any linear transformation between finite-dimensional Hilbert
               spaces is bounded.
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            14.  Prove that if   / i , then ker ² ³  is a closed subspace of  .
                                                                      .
            15.  Prove that a Hilbert space is separable if and only if hdim²/³  L
            16.  Can a Hilbert space have countably infinite Hamel dimension?

            17.  What is the Hamel dimension of M² ³ ?
                                            o
            18.  Let   and   be bounded linear operators on  . Verify the following:

                                                   /

               a )  )) ~       ( () )


               b )  )    )   )  b  )   )      ) b
               c )  )  )    )  )  )   )
                                                           i
            19.  Use the Riesz representation theorem to show that /š /  for any Hilbert
               space .
                    /
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