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Metric Vector Spaces: The Theory of Bilinear Forms  269




                                         %~º h Á %»
            is an isomorphism from   to  =  i  . It follows that for each       =  i   there exists a
                                =
            unique vector %=   for which
                                         # ~ º#Á %»
            for all #=  .…

            The requirement that   be nonsingular is necessary. As a simple example, if  =
                              =
            is  totally  singular, then no nonzero linear functional could possibly be
            represented by an inner product.

            The Riesz representation theorem applies to nonsingular metric vector spaces.
            However, we can also achieve something useful for singular  subspaces   of a
                                                                        :
            nonsingular metric vector space. The reason is that any linear functional   : i
            can be extended to a linear functional   on  , where it has a Riesz vector, that

                                                =
            is,
                                   # ~º#Á 9 » ~º h Á 9 »#


            Hence,   also has this form, where its “Riesz vector” is an element of  , but is

                                                                      =
            not necessarily in  .
                           :
                         (
            Theorem 11.6  The Riesz representation theorem for subspaces)  Let   be a
                                                                        :
            subspace of a metric vector space  . If either   or   is nonsingular, the linear
                                        =
                                                      :
                                                  =
            map  ¢= ¦ : i  defined by
                                        %~º h Á %»O :
            is surjective and has kernel  :  ž  . Hence, for any linear functional       :  i  , there
                                   )
                (
            is a  not necessarily unique  vector %=   for which   ~º Á %»  for all   : .
            Moreover, if   is nonsingular, then   can be taken from  , in which case it is
                                                           :
                       :
                                          %
            unique.…
            Orthogonal Complements and Orthogonal Direct Sums
            Definition A metric vector space  =   is the  orthogonal direct  sum   of  the
            subspaces   and  , written
                     :
                          ;
                                        =~ : p ;
            if =~ : l ;  and : ž ; .…
            If   is a subspace of a real inner product space, the projection theorem says that
              :
            the orthogonal complement :  ž  of   is a true vector space complement of  ,
                                                                           :
                                         :
            that is,
                                       =~ : p :  ž
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