By A M Garsia

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Remark. For a discussion of Vitali's theorem from the point of view of (c), see E. C. Titchmarsh, Theory of Functions, Oxford Univ. Press, London and New York, 1939, pp. 168-170. 1 S. MacLane T h e g e o m e t r y of H i l b e r t s p a c e FINITE-DIMENSIONAL VECTOR SPACES HAVE THREE KINDS OF PROPERTIES WHOSE GENERALIZATIONS WE WL I L STUDY IN THE NEXT FOUR CHAPTERS: LINEAR PROPERTIES, METRIC PROPERTIES, AND GEOMETRIC PROPERTIES. IN THIS CHAPTER WE STUDY VECTOR SPACES THAT HAVE AN INNER PRODUCT, A GENERALIZATION OF THE USUAL DOT PRODUCT ONFINITEDIMENSIONAL VECTOR SPACES.

Proof Suppose Jf is separable and let {x„} be a countable dense set. By throwing out some of the x„'s we can get a subcollection of independent vectors whose span (finite linear combinations) is the same as the { x j and is thus dense. Applying the G r a m - S c h m i d t procedure to this subcollection we obtain a countable complete orthonormal system. 6 that the set of finite linear combinations of the yn with rational coefficients is dense in Jf. Since this set is countable, J f is separable.

F22. Prove thata-rings are closed under countable intersections. 23. (a) Let be a family of substs of M. Prove there is a smallest σ-field & with & <= We say Sf generates & . (b) Let Τ: M-+N where Μ, Ν have associated σ-fields ^ , & . Let Sf generate & . l c St for all S G Sf. Prove Τ is measurable if and only if T~ [S] *24. Let /χ, ν be two finite measures. t. μ if and only if (Υε)(3δ) μ(Α) < δ implies v(A) < ε. 25. 22. 26. Construct a sequence of functions,/,, which are continuous and pointwise convergent to a function/which is not continuous.