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Hilbert Spaces 1. Cauchy-Schwarz-Bunyakowsky (October 29, 2016) Hilbert spaces Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/fun/notes 2016-17/03 hsp.pdf] Hilbert spaces are possibly-infinite-dimensional analogues of the familiar finite-dimensional Euclidean spaces. In particular, Hilbert spaces have inner products, so notions of perpendicularity (or orthogonality), and orthogonal projection are available. Reasonably enough, in the infinite-dimensional case we must be careful not to extrapolate too far based only on the finite-dimensional case. Unfortunately, few naturally-occurring spaces of functions are Hilbert spaces. Given the intuitive geometry of Hilbert spaces, this is disappointing, suggesting that physical intuition is a little distant from the behavior of natural spaces of functions. However, a little later we will see that suitable families of Hilbert spaces can capture what we want. Such ideas were developed by Beppo Levi (1906), Frobenius (1907), and Sobolev (1930's). These ideas do fit into Schwartz' (c. 1950) formulation of his notion of distributions, but it seems that they were not explicitly incorporated, or perhaps were viewed as completely obvious at that point. We will see that Levi-Sobolev ideas offer some useful specifics in addition to Schwartz' over-arching ideas. Most of the geometric results on Hilbert spaces are corollaries of the minimum principle. Most of what is done here applies to vector spaces over either R or C. 1. Cauchy-Schwarz-Bunyakowski inequality 2. Example: `2 3. Completions, infinite sums 4. Minimum principle, orthogonality 5. Parseval equality, Bessel inequality 6. Riemann-Lebesgue lemma 7. Gram-Schmidt process 8. Linear maps, linear functionals, Riesz-Fr´echet theorem 9. Adjoint maps 1. Cauchy-Schwarz-Bunyakowsky inequality A complex vector space V with a complex-valued function h; i : V × V !! C of two variables on V is a (hermitian) inner product space or pre-Hilbert space, and h; i is a (hermitian) inner product, when we have the usual conditions 8 hx; yi = hy; xi (the hermitian-symmetric property) > 0 0 > hx + x ; yi = hx; yi + hx ; yi (additivity in first argument) <> hx; y + y0i = hx; yi + hx; y0i (additivity in second argument) > hx; xi ≥ 0 (and equality only for x = 0: positivity) > > hαx; yi = αhx; yi (linearity in first argument) : hx; αyi =α ¯hx; yi (conjugate-linearity in second argument) Among other easy consequences of these requirements, for all x; y 2 V hx; 0i = h0; yi = 0 where inside the angle-brackets the 0 is the zero-vector, and outside it is the zero-scalar. 1 Paul Garrett: Hilbert spaces (October 29, 2016) The associated norm j j on V is defined by jxj = hx; xi1=2 with the non-negative square-root. Even though we use the same notation for the norm on V as for the usual complex value jj, context will make clear which is meant. The metric on a Hilbert space is d(v; w) = jv − wj: the triangle inequality follows from the Cauchy-Schwarz-Bunyakowsky inequality just below. For two vectors v; w in a pre-Hilbert space, if hv; wi = 0 then v; w are orthogonal or perpendicular, sometimes written v ? w. A vector v is a unit vector if jvj = 1. There are several essential algebraic identities, variously and ambiguously called polarization identities. First, there is jx + yj2 + jx − yj2 = 2jxj2 + 2jyj2 which is obtained simply by expanding the left-hand side and cancelling where opposite signs appear. In a similar vein, jx + yj2 − jx − yj2 = 2hx; yi + 2hy; xi = 4Rehx; yi Therefore, (jx + yj2 − jx − yj2) + i(jx + iyj2 − jx − iyj2) = 4hx; yi These and closely-related identites are of frequent use. [1.1] Theorem: (Cauchy-Schwarz-Bunyakowsky inequality) jhx; yij ≤ jxj · jyj with strict inequality unless x; y are collinear, i.e., unless one of x; y is a multiple of the other. Proof: Suppose that x is not a scalar multiple of y, and that neither x nor y is 0. Then x − αy is not 0 for any complex α. Consider 0 < jx − αyj2 We know that the inequality is indeed strict for all α since x is not a multiple of y. Expanding this, 0 < jxj2 − αhx; yi − α¯hy; xi + αα¯jyj2 Let α = µt with real t and with jµj = 1 so that µhx; yi = jhx; yij Then 0 < jxj2 − 2tjhx; yij + t2jyj2 The minimum of the right-hand side, viewed as a function of the real variable t, occurs when the derivative vanishes, i.e., when 0 = −2jhx; yij + 2tjyj2 Using this minimization as a mnemonic for the value of t to substitute, we indeed substitute jhx; yij t = jyj2 into the inequality to obtain jhx; yij2 jhx; yij 0 < jxj2 + · jyj2 − 2 · jhx; yij jyj2 jyj2 2 Paul Garrett: Hilbert spaces (October 29, 2016) which simplifies to jhx; yij2 < jxj2 · jyj2 as desired. === [1.2] Corollary: (Triangle inequality) For v; w in a Hilbert space V , we have jv + wj ≤ jvj + jwj. Thus, with distance function d(v; w) = jv − wj, we have the triangle inequality d(x; z) = jx − zj = j(x − y) + (y − z)j ≤ jx − yj + jy − zj = d(x; y) + d(y; z) Proof: Squaring and expanding, noting that hv; wi + hw; vi = 2Rehv; wi, (jvj + jwj)2 − jv + wj2 = jvj2 + 2jvj · jwj + jwj2 − jvj2 + hv; wi + hw; vi + jwj2 ≥ 2jvj · jwj − 2jhv; wij ≥ 0 giving the asserted inequality. === An inner product space complete with respect to the metric arising from its inner product (and norm) is a Hilbert space. [1.3] Continuity issues The map h; i : V × V −! C is continuous as a function of two variables. Indeed, suppose that jx − x0j < " and jy − y0j < " for x; x0; y; y0 2 V . Then hx; yi − hx0; y0i = hx − x0; yi + hx0; yi − hx0; y0i = hx − x0; yi + hx0; y − y0i Using the triangle inequality for the ordinary absolute value, and then the Cauchy-Schwarz-Bunyakowsky inequality, we obtain jhx; yi − hx0; y0ij ≤ jhx − x0; yij + jhx0; y − y0ij ≤ jx − x0jjyj + jx0jjy − y0j < "(jyj + jx0j) This proves the continuity of the inner product. Further, scalar multiplication and vector addition are readily seen to be continuous. In particular, it is easy to check that for any fixed y 2 V and for any fixed λ 2 C× both maps x ! x + y x ! λx are homeomorphisms of V to itself. 3 Paul Garrett: Hilbert spaces (October 29, 2016) 2. Example: `2 Before further abstract discussion, we note that, up to isomorphism, there is essentially just one infinite- dimensional Hilbert space occurring in practice, namely the space `2 constructed as follows. Most infinite- dimensional Hilbert spaces occurring in practice have a countable dense subset, because the Hilbert spaces are completions of spaces of continuous functions on topological spaces with a countably-based topology. Lest anyone be fooled, often subtlety is in the description of the isomorphisms and mappings among such Hilbert spaces. Let `2 be the collection of sequences f = ff(i) : 1 ≤ i < 1g of complex numbers meeting the constraint 1 X jf(i)j2 < +1 i=1 For two such sequences f and g, the inner product is X hf; gi = f(i)g(i) i [2.1] Claim: `2 is a vector space. The sum defining the inner product on `2 is absolutely convergent. Proof: That `2 is closed under scalar multiplication is clear. For f; g 2 `2, by Cauchy-Schwarz-Bunyakowsky, 1 1 2 2 X X X 2 X 2 f(i) · g(i) ≤ jf(i)j · jg(i)j ≤ jf(i)j · jg(i)j ≤ jfj`2 · jgj`2 n≤N n≤N n≤N n≤N giving the absolute convergence of the infinite sum for hf; gi. Then, expanding, X X jf(i) + g(i)j2 ≤ jf(i)j2 + 2jf(i)j · jg(i)j + jg(i)j2 < +1 n≤N n≤N by the previous. === [2.2] Claim: `2 is complete. 2 Proof: Let ffng be a Cauchy sequence of elements in ` . For every i 2 f1; 2; 3;:::g, 2 X 2 2 jfm(i) − fn(i)j ≤ jfm(i) − fn(i)j = jfm − fnj`2 i≥1 2 so f(i) = limn fn(i) exists for every i, and is the obvious candidate for the limit in ` . It remains to see that this limit is indeed in `2. This will follow from an easy case of Fatou's lemma: X 2 X X X 2 jf(i)j = j lim fn(i)j = j lim inf fn(i)j ≤ lim inf jfn(i)j = lim inf jfnj 2 n n n n ` i i i i 2 Since ffng is a Cauchy sequence, certainly limn jfnj`2 exists. === [2.3] Remark: A similar result holds for L2(X; µ) for general measure spaces X; µ, but requires more preparation. 4 Paul Garrett: Hilbert spaces (October 29, 2016) 3. Completions, infinite sums An arbitrary pre-Hilbert space can be completed as metric space, giving a Hilbert space. Since metric spaces 1 1 1 have countable local bases for their topology (e.g., open balls of radii 1; 2 ; 3 ; 4 ;:::) all points in the completion are limits of Cauchy sequences (rather than being limits of more complicated Cauchy nets). The completion inherits an inner product defined by a limiting process hlim xm; lim yni = limhxm; yni m n m;n It is not hard to verify that the indicated limit exists (for Cauchy sequences fxmg; fyng), and gives a hermitian inner product on the completion. The completion process does nothing to a space which is already complete. In a Hilbert space, we can consider infinite sums X vα α2A for sets fvα : α 2 Ag of vectors in V .
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