# Download Applications of Functional Analysis and Operator Theory by V. Hutson PDF

By V. Hutson

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Consider the Lebesgue measure of the set S of rationals in an interval X of R The measure of a single point is zero, and S is a countable union of single points. 6(ii), fJ(S) = O. By taking complements, we deduce that the measure of the irrationals is the length of X. 15 Example. Another useful measure may be obtained from the theorem. Let y be any non-decreasing function on IR which is continuous on the right. For a, b E IR define y«a, bJ) = y(b) - yea), y(a, b)) = lim y(x) - yea), x--'>h- with similar definitions for the other types of interval.

15. (iii) => (iv). 4). (iv) => (i). 9)is zero, whencef = 0. 18 Theorem. A separable Hilbert space has an orthonormal basis. 21), rejecting at each stage the nex t j ~ if it together with the previous orthonormal elements are not linearly independent. x] = Yf. ff is an orthonormal basis. 19 Example. 8). Let ell be the element whose mth component is bl1l n - e s s e n t i a l l y ell is a unit vector along 34 BANACH SPACES the "nth axis". Clearly {eJ is an orthonormal set. Now ifI = (/1'/2"") is any vector in t 2' then (f, e) = In' Therefore I = 0 if (f, en) = 0 for n ~ 1.

Thus f1* is not a measure (although it is called an outer measure i). The strategy is to pick out a class of sets on which f1* is additive-and so a measure. This is by no means easy, but the class turns out to consist of all sets Y which satisfy f1*(S rl y) + f1*(S\Y) = f1*(S) for every S E Y'. These sets form the a-algebra of the theorem. We have now reached our goal. 11 and extend this as in the previous theorem. 13 Definition. The o-aigebra produced from the length of intervals, area of rectangles, ...