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Inner-Product Spaces, Euclidean Spaces Chapter 4 Inner-Product Spaces, Euclidean Spaces As in Chap.2, the term “linear space” will be used as a shorthand for “finite dimensional linear space over R”. However, the definitions of an inner-product space and a Euclidean space do not really require finite- dimensionality. Many of the results, for example the Inner-Product In- equality and the Theorem on Subadditivity of Magnitude, remain valid for infinite-dimensional spaces. Other results extend to infinite-dimensional spaces after suitable modification. 41 Inner-Product Spaces Definition 1: An inner-product space is a linear space endowed with V additional structure by the prescription of a non-degenerate quadratic form sq Qu( ) (see Sect.27). The form sq is then called the inner square of ∈ V and the corresponding symmetric bilinear form V ip := sq Sym ( 2, R) = Sym( , ∗) ∈ 2 V ∼ V V the inner product of . V We say that the inner-product space is genuine if sq is strictly posi- V tive. It is customary to use the following simplified notations: v·2 := sq(v) when v , (41.1) ∈ V u v := ip(u, v) = (ip u)v when u, v . (41.2) · ∈ V 133 134 CHAPTER 4. INNER-PRODUCT SPACES, EUCLIDEAN SPACES The symmetry and bilinearity of ip is then reflected in the following rules, valid for all u, v, w and ξ R ∈ V ∈ u v = v u, (41.3) · · w (u + v) = w u + w v, (41.4) · · · u (ξv) = ξ(u v)=(ξu) v (41.5) · · · We say that u is orthogonal to v if u v = 0. The assumption ∈ V ∈ V · that the inner product is non-degenerate is expressed by the statement that, given u , ∈ V (u v =0 for all v ) = u = 0. (41.6) · ∈ V ⇒ In words, the zero of is the only member of that is orthogonal to every V V member of . V Since ip Lin( , ∗) is injective and since dim = dim ∗, it follows ∈ V V V V from the Pigeonhole Principle for Linear Mappings that ip is a linear iso- morphism. It induces the linear isomorphism ip⊤ : ∗∗ ∗. Since V → V ip is symmetric, we have ip⊤ = ip when ∗∗ is identified with as ex- V V plained in Sect.22. Thus, this identification is the same as the isomorphism (ip⊤)−1ip : ∗∗ induced by ip . Therefore, there is no conflict if we V → V use ip to identify with ∗. V V From now on we shall identify = ∗ by means of ip except that, given V ∼ V v , we shall write v := ip v for the corresponding element in ∗ so as ∈ V · V to be consistent with the notation (41.2). Every space RI of families of numbers, indexed on a given finite set I, carries the natural structure of an inner-product space whose inner square is given by ·2 2 2 sq(λ) = λ := λ = λi (41.7) X Xi∈I for all λ RI . The corresponding inner product is given by ∈ (ip λ)µ = λ µ = λµ = λiµi (41.8) · X Xi∈I for all λ, µ RI . The identification (RI )∗ = RI resulting from this inner ∈ ∼ product is the same as the one described in Sect.23. Let and be inner-product spaces. The identifications ∗ = and V W V ∼ V ∗ = give rise to the further identifications such as W ∼ W Lin( , ) = Lin( , ∗) = Lin ( , R), V W ∼ V W ∼ 2 V×W 41. INNER-PRODUCT SPACES 135 Lin( , ) = Lin( ∗, ∗). W V ∼ W V Thus L Lin( , ) becomes identified with the bilinear form ∈ V W L Lin ( , R) whose value at (v, w) satisfies ∈ 2 V×W ∈V×W L(v, w)=(Lv) w, (41.9) · and L⊤ Lin( ∗, ∗) becomes identified with L⊤ Lin( , ) = ∈ W V ∈ W V ∼ Lin ( , R) in such a way that 2 W × V (L⊤w) v = L⊤(w, v) = L(v, w)=(Lv) w (41.10) · · for all v , w . ∈ V ∈W We identify the supplementary subspaces Sym ( 2, R) and Skew ( 2, R) 2 V 2 V of Lin ( 2, R) = Lin (see Prop.7 of Sect.24) with the supplementary sub- 2 V ∼ V spaces Sym := S Lin S = S⊤ , (41.11) V { ∈ V | } Skew := A Lin A⊤ = A (41.12) V { ∈ V | − } of Lin . The members of Sym are called symmetric lineons and the V V members of Skew skew lineons. V Given L Lin( , ) and hence L⊤ Lin( , ) we have L⊤L Sym ∈ V W ∈ W V ∈ V and LL⊤ Sym . Clearly, if L⊤L is invertible (and hence injective), then ∈ W L must also be injective. Also, if L is invertible, so is L⊤L. Applying these observations to the linear-combination mapping of a family f := (fi i I) | ∈ in and noting the identification V ⊤ 2 I2 I Gf := lncf lncf =(fi fk (i,k) I ) R = Lin(R ) (41.13) · | ∈ ∈ ∼ we obtain the following results. Proposition 1: Let f := (fi i I) be a family in . Then f is linearly | ∈ V independent if the matrix Gf given by (41.13) is invertible. Proposition 2: A family b := (bi i I) in is a basis if and only if | ∈ V the matrix Gb is invertible and ♯I = dim . V ∗ Let b := (bi i I) be a basis of . The identification = identifies | ∈ V V ∼ V the dual of the basis b with another basis b∗ of in such a way that V ∗ b bk = δi,k for all i,k I, (41.14) i · ∈ where δi,k is defined by (16.2). Using the notation (41.13) we have ∗ bk = (Gb)k,ibi (41.15) Xi∈I 136 CHAPTER 4. INNER-PRODUCT SPACES, EUCLIDEAN SPACES and −1 Gb∗ = Gb . (41.16) Moreover, for each v , we have ∈ V lnc−1(v) = b∗ v := (b∗ v i I) (41.17) b · i · | ∈ and hence ∗ v = lncb(b v) = lncb∗ (b v). (41.18) · · For all u, v , we have ∈ V ∗ ∗ u v =(b u) (b v) = (bi u)(b v). (41.19) · · · · · i · Xi∈I We say that a family e := (ei i I) in is orthonormal if | ∈ V 1 or 1 if i = k ei ek = − for all i,k I (41.20) · 0 if i = k ∈ 6 ·2 We say that e is genuinely orthonormal if, in addition, ei = 1 for all i I, which—using the notation (16.2)—means that ∈ ei ek = δi,k for all i,k I. (41.21) · ∈ To say that e is orthonormal means that the matrix Ge, as defined by (41.13), is diagonal and each of its diagonal terms is 1 or 1. It is − clear from Prop.1 that every orthonormal family is linearly independent and from Prop.2 that a given orthonormal family is a basis if and only if ♯I = dim . Comparing (41.21) with (41.14), we see that a basis b is V genuinely orthonormal if and only if it coincides with its dual b∗. The standard basis δI := (δI i I) (see Sect.16) is a genuinely or- i | ∈ thonormal basis of RI . Let v , w be given. Then w v Lin( ∗, ) becomes identified ∈ V ∈W ⊗ ∈ V W with the element w v of Lin( , ) whose values are given by ⊗ V W (w v)u =(v u)w for all u . (41.22) ⊗ · ∈ V We say that a subspace of a given inner-product space is regular U V if the restriction sq of the inner square to is non-degenerate, i.e., if for |U U each u , ∈U (u v =0 for all v ) = u = 0. (41.23) · ∈U ⇒ 41. INNER-PRODUCT SPACES 137 If is regular, then sq endows with the structure of an inner-product U |U U space. If is not regular, it does not have a natural structure of an inner- U product space. The identification ∗ = identifies the annihilator ⊥ of a given subset V ∼ V S of with the subspace S V ⊥ = v v u =0 for all u S { ∈V| · ∈ S} of . Thus, ⊥ consists of all elements of that are orthogonal to every V S V element of . The following result is very easy to prove with the use of the S Formula for Dimension of Annihilators (Sect.21) and of Prop.5 of Sect.17. Characterization of Regular Subspaces: Let be a subspace of . U V Then the following are equivalent: (i) is regular, U (ii) ⊥ = 0 , U∩U { } (iii) + ⊥ = , U U V (iv) and ⊥ are supplementary. U U Moreover, if is regular, so is ⊥. U U If is regular, then its annihilator ⊥ is also called the orthogonal U U supplement of . The following two results exhibit a natural one-to-one U correspondence between the regular subspaces of and the symmetric idem- V potents in Lin , i.e., the lineons E Lin that satisfy E = E⊤ and E2 = E. V ∈ V Proposition 3: Let E Lin be an idempotent. Then E is symmetric ∈ V if and only if Null E = (Rng E)⊥. Proof: If E = E⊤, then Null E = (Rng E)⊥ follows from (21.13). Assume that Null E = (Rng E)⊥. It follows from (21.13) that Null E = Null E⊤ and hence from (22.9) that Rng E⊤ = Rng E. Since, by (21.6), (E⊤)2 = (E2)⊤ = E⊤, it follows that E⊤ is also an idempotent. The assertion of uniqueness in Prop.4 of Sect.19 shows that E = E⊤. Proposition 4: If E Lin is symmetric and idempotent, then Rng E ∈ V is a regular subspace of and Null E is its orthogonal supplement. Con- V versely, if is a regular subspace of , then there is exactly one symmetric U V idempotent E Lin such that = Rng E.
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