By Jerrold Marsden, Alan Weinstein

This is often the second one booklet of a three-volume paintings known as "Calculus" via Jerrold Marsden and Alan Weinstein. This ebook is the outgrowth of the authors' adventure instructing calculus at Berkeley. It covers thoughts and functions of integration, endless sequence, and differential equations. through the e-book, the authors inspire the learn of calculus utilizing its purposes. Many solved difficulties are incorporated, and huge routines are given on the finish of every part. moreover, a separate scholar consultant has been ready.

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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, ...