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Hilbert Spaces C H A P T E R 12 Hilbert Spaces The material to be presented in this chapter is essential for all advanced work in physics and analysis. We have attempted to present several relatively diffi- cult theorems in sufficient detail that they are readily understandable by readers with less background than normally might be required for such results. However, we assume that the reader is quite familiar with the contents of Appendices A and B, and we will frequently refer to results from these appendices. Essentially, this chapter serves as an introduction to the theory of infinite-dimensional vector spaces. Throughout this chapter we let E, F and G denote normed vector spaces over the real or complex number fields only. 12.1 MATHEMATICAL PRELIMINARIES This rather long first section presents the elementary properties of limits and continuous functions. While most of this material properly falls under the heading of analysis, we do not assume that the reader has already had such a course. However, if these topics are familiar, then the reader should briefly scan the theorems of this section now, and return only for details if and when it becomes necessary. For ease of reference, we briefly repeat some of our earlier definitions and results. By a norm on a vector space E, we mean a mapping ˜ ˜: E ‘ ® satis- fying: 619 620 HILBERT SPACES (N1) ˜u˜ ˘ 0 for every u ∞ E and ˜u˜ = 0 if and only if u = 0 (positive definiteness). (N2) ˜cu˜ = \c\ ˜u˜ for every u ∞ E and c ∞ F. (N3) ˜u + v˜ ¯ ˜u˜ + ˜v˜ (triangle inequality). If there is more than one norm on E under consideration, then we may denote them by subscripts such as ˜ ˜ì etc. Similarly, if we are discussing more than one space, then the norm associated with a space E will sometimes be denoted by ˜ ˜E . We call the pair (E, ˜ ˜) a normed vector space. If E is a complex vector space, we define the Hermitian inner product as the mapping Ó , Ô: E ª E ‘ ç such that for all u, v, w ∞ E and c ∞ ç we have: (IP1) Óu, v + wÔ = Óu, vÔ + Óu, wÔ. (IP2) Ócu, vÔ = c*Óu, vÔ. (IP3) Óu, vÔ = Óv, uÔ*. (IP4) Óu, uÔ ˘ 0 and Óu, uÔ = 0 if and only if u = 0. where * denotes complex conjugation. A Hermitian inner product is some- times called a sesquilinear form. Note that (IP2) and (IP3) imply Óu, cvÔ = Ócv, uÔ* = cÓv, uÔ* = cÓu, vÔ and that (IP3) implies Óv, vÔ is real. As usual, if Óu, vÔ = 0 we say that u and v are orthogonal, and we some- times write this as u ‡ v. If we let S be a subset of E, then the collection {u ∞ E: Óu, vÔ = 0 for every v ∞ S} is a subspace of E called the orthogonal complement of S, and is denoted by SÊ. It should be remarked that many authors define Ócu, vÔ = cÓu, vÔ rather than our (IP2), and the reader must be careful to note which definition is being fol- lowed. Furthermore, there is no reason why we could not have defined a map- ping E ª E ‘ ®, and in this case we have simply an inner product on E (where obviously there is now no complex conjugation). The most common example of a Hermitian inner product is the standard inner product on çn = ç ª ~ ~ ~ ª ç defined for all x = (xè, . , xñ) and y = (yè, . , yñ) in çn by n Óx,!yÔ = !xi *yi !!. i=1 12.1 MATHEMATICAL PRELIMINARIES 621 We leave it to the reader to verify conditions (IP1) - (IP4). Before defining a norm on çn, we again prove (in a slightly different manner from that in Chapter 2) the Cauchy-Schwartz inequality. Example 12.1 Let E be a complex (or real) inner product space, and let u, v ∞ E be nonzero vectors. Then for any a, b ∞ ç we have 0 ¯ Óau + bv, au + bvÔ = \a\2Óu, uÔ + a*bÓu, vÔ + b*aÓv, uÔ + \b\2Óv, vÔ . Now note that the middle two terms are complex conjugates of each other, and hence their sum is 2Re(a*bÓu, vÔ). Therefore, letting a = Óv, vÔ and b = -Óv, uÔ, we have 0 ¯ Óv, vÔ2Óu, uÔ - 2Óv, vÔ\Óu, vÔ\2 + \Óu, vÔ\2Óv, vÔ which is equivalent to Óv, vÔ\Óu, vÔ\2 ¯ Óv, vÔ2Óu, uÔ . Since v ≠ 0 we have Óv, vÔ ≠ 0, and hence dividing by Óv, vÔ and taking the square root yields the desired result \Óu, vÔ\ ¯ Óu, uÔ1/2 Óv, vÔ1/2 . ∆ If a vector space E has an inner product defined on it, then we may define a norm on E by ˜v˜ = Óv, vÔ1/2 for all v ∞ E. Properties (N1) and (N2) for this norm are obvious, and (N3) now follows from the Cauchy-Schwartz inequality and the fact that ReÓu, vÔ ¯ \Óu, vÔ\: ˜u + v˜2 = Óu + v,!u + vÔ = ˜u˜2+2ReÓu,!vÔ+ ˜v˜2 ! ˜u˜2+2|Óu,!vÔ| + ˜v˜2 ! ˜u˜2+ 2˜u˜˜v˜ + ˜v˜2 ( u v )2 !!. = ˜ ˜ + ˜ ˜ We leave it to the reader (Exercise 12.1.1) to prove the so-called parallel- ogram law in an inner product space (E, Ó , Ô): ˜u + v˜2 + ˜u - v˜2 = 2˜u˜2 + 2˜v˜2 . 622 HILBERT SPACES The geometric meaning of this formula in ®2 is that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of the sides. If Óu, vÔ = 0, then the reader can also easily prove the Pythagorean theorem: ˜u + v˜2 = ˜u˜2 + ˜v˜2 if u ‡ v . In terms of the standard inner product on çn, we now define a norm on çn by n 2 2 ˜x˜ = Óx,!xÔ = !|xi | !!. i=1 The above results now show that this does indeed satisfy the requirements of a norm. Continuing, if (E, ˜ ˜) is a normed space, then we may make E into a metric space (E, d) by defining d(u, v) = ˜u - v˜ . Again, the only part of the definition of a metric space (see Appendix A) that is not obvious is (M4), and this now follows from (N3) because d(u,!v) = ˜u ! v˜ = ˜u ! w + w ! v˜ " ˜u ! w˜ + ˜w ! v˜ = d(u,!w)+ d(w,!v)!!. The important point to get from all this is that normed vector spaces form a special class of metric spaces. This means that all the results from Appendix A and many of the results from Appendix B will carry over to the case of normed spaces. In Appendix B we presented the theory of sequences and series of numbers. As we explained there however, many of the results are valid as well for normed vector spaces if we simply replace the absolute value by the norm. For example, suppose A ™ E and let v ∞ E. Recall that v is said to be an accumulation point of A if every open ball centered at v contains a point of A distinct from v. In other words, given ´ > 0 there exists u ∞ A, u ≠ v, such that ˜u - v˜ < ´. As expected, if {vñ} is a sequence of vectors in E, then we say that {vñ} converges to the limit v ∞ E if given ´ > 0, there exists an integer N > 0 such that n ˘ N implies ˜vñ - v˜ < ´. As usual, we write lim vñ = limn‘Ÿvñ = v. If there exists a neighborhood of v (i.e., an open ball contain- ing v) such that v is the only point of A in this neighborhood, then we say that v is an isolated point of A. 12.1 MATHEMATICAL PRELIMINARIES 623 Example 12.2 Suppose lim vñ = v. Then for every ´ > 0, there exists N such that n ˘ N implies ˜v - vñ˜ < ´. From Example 2.11 we then see that \ ˜v˜ - ˜vñ˜ \ ¯ ˜v - vñ˜ < ´ and hence directly from the definition of lim ˜vñ˜ we have ˜ lim vñ ˜ = ˜v˜ = lim ˜vñ˜ . This result will be quite useful in several later proofs. ∆ Note that if v is an accumulation point of A, then for every n > 0 there exists vñ ∞ A, vn ≠ v, such that ˜vñ - v˜ < 1/n. In particular, for any ´ > 0, choose N so that 1/N < ´. Then for all n ˘ N we have ˜vñ - v˜ < 1/n < ´ so that {vñ} converges to v. Conversely, it is clear that if vñ ‘ v with vñ ∞ A, vn ≠ v, then v must necessarily be an accumulation point of A. This proves the fol- lowing result. Theorem 12.1 If A ™ E, then v is an accumulation point of A if and only if it is the limit of some sequence in A - {v}. A function f: (X, dX) ‘ (Y, dY) is said to be continuous at xà ∞ X if for each ´ > 0 there exists ∂ > 0 such that dX(x, xà) < ∂ implies dY(f(x), f(xà)) < ´. Note though, that for any given ´, the ∂ required will in general be different for each point xà chosen. If f is continuous at each point of X, then we say that f is “continuous on X.” A function f as defined above is said to be uniformly continuous on X if for each ´ > 0, there exists ∂ > 0 such that for all x, y ∞ X with dX(x, y) < ∂ we have dY(f(x), f(y)) < ´. The important difference between continuity and uniform continuity is that for a uniformly continuous function, once ´ is chosen, there is a single ∂ (which will generally still depend on ´) such that this definition applies to all points x, y ∞ X subject only to the requirement that dX(x, y) < ∂.
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