Chapter 8 Basics of Hermitian Geometry 8.1 Sesquilinear Forms, Hermitian Forms, Hermitian Spaces, Pre-Hilbert Spaces In this chapter, we generalize the basic results of Eu- clidean geometry presented in Chapter 6 to vector spaces over the complex numbers. Some complications arise, due to complex conjugation. Recall that for any complex number z C,ifz = x + iy 2 where x, y R,welet z = x,therealpartofz,and z = y,theimaginarypartof2 < z. = We also denote the conjugate of z = x+iy as z = x iy, and the absolute value (or length, or modulus) of z as− z . Recall that z 2 = zz = x2 + y2. | | | | 479 480 CHAPTER 8. BASICS OF HERMITIAN GEOMETRY There are many natural situations where a map ': E E C is linear in its first argument and only semilinear⇥ in! its second argument. For example, the natural inner product to deal with func- tions f : R C,especiallyFourierseries,is ! ⇡ f,g = f(x)g(x)dx, h i ⇡ Z− which is semilinear (but not linear) in g. Definition 8.1. Given two vector spaces E and F over the complex field C,afunctionf : E F is semilinear if ! f(u + v)=f(u)+f(v), f(λu)=λf(u), for all u, v E and all λ C. 2 2 8.1. SESQUILINEAR FORMS, HERMITIAN FORMS 481 Remark:Insteadofdefiningsemilinearmaps,wecould have defined the vector space E as the vector space with the same carrier set E,whoseadditionisthesameasthat of E,butwhosemultiplicationbyacomplexnumberis given by (λ, u) λu. 7! Then, it is easy to check that a function f : E C is ! semilinear i↵ f : E C is linear. ! We can now define sesquilinear forms and Hermitian forms. 482 CHAPTER 8. BASICS OF HERMITIAN GEOMETRY Definition 8.2. Given a complex vector space E,afunc- tion ': E E C is a sesquilinear form i↵it is linear in its first⇥ argument! and semilinear in its second argu- ment, which means that '(u1 + u2,v)='(u1,v)+'(u2,v), '(u, v1 + v2)='(u, v1)+'(u, v2), '(λu, v)='(u, v), '(u, µv)=µ'(u, v), for all u, v, u1,u2, v1,v2 E,andallλ, µ C.A 2 2 function ': E E C is a Hermitian form i↵it is sesquilinear and⇥ if ! '(v, u)='(u, v) for all all u, v E. 2 Obviously, '(0,v)='(u, 0) = 0. 8.1. SESQUILINEAR FORMS, HERMITIAN FORMS 483 Also note that if ': E E C is sesquilinear, we have ⇥ ! '(λu + µv, λu + µv)= λ 2'(u, u)+λµ'(u, v) | | + λµ'(v, u)+ µ 2'(v, v), | | and if ': E E C is Hermitian, we have ⇥ ! '(λu + µv, λu + µv) = λ 2'(u, u)+2 (λµ'(u, v)) + µ 2'(v, v). | | < | | Note that restricted to real coefficients, a sesquilinear form is bilinear (we sometimes say R-bilinear). The function Φ: E C defined such that Φ(u)='(u, u) for all u E is called! the quadratic form associated with '. 2 484 CHAPTER 8. BASICS OF HERMITIAN GEOMETRY The standard example of a Hermitian form on Cn is the map ' defined such that '((x ,...,x ), (y ,...,y )) = x y + x y + + x y . 1 n 1 n 1 1 2 2 ··· n n This map is also positive definite, but before dealing with these issues, we show the following useful proposition. Proposition 8.1. Given a complex vector space E, the following properties hold: (1) A sesquilinear form ': E E C is a Hermitian ⇥ ! form i↵ '(u, u) R for all u E. 2 2 (2) If ': E E C is a sesquilinear form, then ⇥ ! 4'(u, v)='(u + v, u + v) '(u v, u v) − − − + i'(u + iv, u + iv) i'(u iv, u iv), − − − and 2'(u, v)=(1+i)('(u, u)+'(v, v)) '(u v, u v) i'(u iv, u iv). − − − − − − These are called polarization identities. 8.1. SESQUILINEAR FORMS, HERMITIAN FORMS 485 Proposition 8.1 shows that a sesquilinear form is com- pletely determined by the quadratic form Φ(u)='(u, u), even if ' is not Hermitian. This is false for a real bilinear form, unless it is symmetric. For example, the bilinear form ': R2 R2 R defined such that ⇥ ! '((x ,y ), (x ,y )) = x y x y 1 1 2 2 1 2 − 2 1 is not identically zero, and yet, it is null on the diagonal. However, a real symmetric bilinear form is indeed deter- mined by its values on the diagonal, as we saw in Chapter 6. As in the Euclidean case, Hermitian forms for which '(u, u) 0playanimportantrole. ≥ 486 CHAPTER 8. BASICS OF HERMITIAN GEOMETRY Definition 8.3. Given a complex vector space E,aHer- mitian form ': E E C is positive i↵ '(u, u) 0 for all u E,andpositive⇥ ! definite i↵ '(u, u) > 0forall≥ u =0.Apair2 E,' where E is a complex vector space and6 ' is a Hermitianh i form on E is called a pre-Hilbert space if ' is positive, and a Hermitian (or unitary) space if ' is positive definite. We warn our readers that some authors, such as Lang [22], define a pre-Hilbert space as what we define to be a Hermitian space. We prefer following the terminology used in Schwartz [26] and Bourbaki [7]. The quantity '(u, v)isusuallycalledtheHermitian prod- uct of u and v.Wewilloccasionallycallittheinner product of u and v. 8.1. SESQUILINEAR FORMS, HERMITIAN FORMS 487 Given a pre-Hilbert space E,' ,asinthecaseofaEu- clidean space, we also denoteh '(iu, v)as u v, or u, v , or (u v), · h i | and Φ(u)as u . k k p Example 1. The complex vector space Cn under the Hermitian form '((x ,...,x ), (y ,...,y )) = x y + x y + + x y 1 n 1 n 1 1 2 2 ··· n n is a Hermitian space. Example 2. Let l2 denote the set of all countably in- finite sequences x =(xi)i N of complex numbers such 2 2 n 2 that i1=0 xi is defined (i.e. the sequence i=0 xi converges as| n| ). | | P !1 P It can be shown that the map ': l2 l2 C defined such that ⇥ ! 1 ' ((xi)i N, (yi)i N)= xiyi 2 2 i=0 X is well defined, and l2 is a Hermitian space under '.Ac- tually, l2 is even a Hilbert space. 488 CHAPTER 8. BASICS OF HERMITIAN GEOMETRY Example 3. Consider the set [a, b]ofpiecewisebounded Cpiece continuous functions f :[a, b] C under the Hermitian form ! b f,g = f(x)g(x)dx. h i Za It is easy to check that this Hermitian form is positive, but it is not definite. Thus, under this Hermitian form, [a, b]isonlyapre-Hilbertspace. Cpiece Example 4. Let [a, b]bethesetofcomplex-valued C continuous functions f :[a, b] C under the Hermitian form ! b f,g = f(x)g(x)dx. h i Za It is easy to check that this Hermitian form is positive definite. Thus, [a, b]isaHermitianspace. C The Cauchy-Schwarz inequality and the Minkowski in- equalities extend to pre-Hilbert spaces and to Hermitian spaces. 8.1. SESQUILINEAR FORMS, HERMITIAN FORMS 489 Proposition 8.2. Let E,' be a pre-Hilbert space with associated quadratich formi Φ. For all u, v E, we have the Cauchy-Schwarz inequality: 2 '(u, v) Φ(u) Φ(v). | | Furthermore, if E,' is ap Hermitianp space, the equal- ity holds i↵ u andh v arei linearly dependent. We also have the Minkovski inequality: Φ(u + v) Φ(u)+ Φ(v). Furthermore,p if E,' is ap Hermitianp space, the equal- ity holds i↵ u andh v arei linearly dependent, where in addition, if u =0and v =0, then u = λv for some real λ such that6 λ>0. 6 As in the Euclidean case, if E,' is a Hermitian space, the Minkovski inequality h i Φ(u + v) Φ(u)+ Φ(v) shows that thep map u pΦ(u)isapnorm on E. 7! p 490 CHAPTER 8. BASICS OF HERMITIAN GEOMETRY The norm induced by ' is called the Hermitian norm induced by '. We usually denote Φ(u)as u ,andtheCauchy-Schwarz inequality is written as k k p u v u v . | · |k kk k Since a Hermitian space is a normed vector space, it is atopologicalspaceunderthetopologyinducedbythe norm (a basis for this topology is given by the open balls B0(u, ⇢)ofcenteru and radius ⇢>0, where B (u, ⇢)= v E v u <⇢ . 0 { 2 |k − k } If E has finite dimension, every linear map is continuous. 8.1. SESQUILINEAR FORMS, HERMITIAN FORMS 491 The Cauchy-Schwarz inequality u v u v | · |k kk k shows that ': E E C is continuous, and thus, that is continuous.⇥ ! kk If E,' is only pre-Hilbertian, u is called a semi- normh . i k k In this case, the condition u =0 implies u =0 k k is not necessarily true. However, the Cauchy-Schwarz inequality shows that if u =0,thenu v =0forallv E. k k · 2 We will now basically mirror the presentation of Euclidean geometry given in Chapter 6 rather quickly, leaving out most proofs, except when they need to be seriously amended. 492 CHAPTER 8. BASICS OF HERMITIAN GEOMETRY 8.2 Orthogonality, Duality, Adjoint of A Linear Map In this section, we assume that we are dealing with Her- mitian spaces. We denote the Hermitian inner product as u v or u, v . · h i The concepts of orthogonality, orthogonal family of vec- tors, orthonormal family of vectors, and orthogonal com- plement of a set of vectors, are unchanged from the Eu- clidean case (Definition 6.2). For example, the set [ ⇡,⇡]ofcontinuousfunctions C − f :[ ⇡,⇡] C is a Hermitian space under the product − ! ⇡ f,g = f(x)g(x)dx, h i ⇡ Z− ikx and the family (e )k Z is orthogonal.