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292              Chapter 5                    Norms, Inner Products, and Orthogonality

                                    Clearly, 2 (p+2)/p  =4 only when p =2. Details for the ∞-norm are asked for
                                    in Exercise 5.3.7.
                                    Conclusion: For applications that are best analyzed in the context of an inner-
                                    product space (e.g., least squares problems), we are limited to the euclidean
                                    norm or else to one of its variation such as the elliptical norm in (5.3.5).

                                                                                  n     n
                                        Virtually all important statements concerning    or C  with the standard
                                    inner product remain valid for general inner-product spaces—e.g., consider the
                                    statement and proof of the general CBS inequality. Advanced or more theoretical
                                    texts prefer a development in terms of general inner-product spaces. However,
                                                                                            n       n
                                    the focus of this text is matrices and the coordinate spaces    and C , so
                                                                                         n     n
                                    subsequent discussions will usually be phrased in terms of    or C  and their
                                    standard inner products. But remember that extensions to more general inner-
                                    product spaces are always lurking in the background, and we will not hesitate
                                    to use these generalities or general inner-product notation when they serve our
                                    purpose.
                   Exercises for section 5.3



                                                    x 1         y 1
                                    5.3.1. For x =  x 2 , y =   y 2 , determine which of the following are inner
                                                    x 3         y 3
                                                        3×1
                                           products for    .
                                              (a)   x y  = x 1 y 1 + x 3 y 3 ,
                                              (b)   x y  = x 1 y 1 − x 2 y 2 + x 3 y 3 ,
                                              (c)   x y  =2x 1 y 1 + x 2 y 2 +4x 3 y 3 ,
                                                                 2 2
                                                                        2 2
                                                           2 2
                                              (d)   x y  = x y + x y + x y .
                                                           1 1   2 2    3 3
                                    5.3.2. Fora general inner-product space V, explain why each of the following
                                           statements must be true.
                                              (a) If  x y  =0 for all x ∈V, then y = 0.
                                              (b)   αx y  = α  x y  for all x, y ∈V and for all scalars α.
                                              (c)   x + y z  =  x z  +  y z  for all x, y, z ∈V.


                                    5.3.3. Let V be an inner-product space with an inner product  x y  . Explain

                                           why the function defined by     =        satisfies the first two norm
                                           properties in (5.2.3) on p. 280.

                                                                              2
                                    5.3.4. Fora real inner-product space with     =       , derive the inequality
                                                               2      2
                                                             x  +  y
                                                     x y ≤             .   Hint: Consider x − y.
                                                                 2
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