Lethn(x) :=nforx E (0, I/n)andhn(x) :=Oelsewherein[O, 1]. Does there exist h E [0,1]such

Show that Hake's Theorem 10.2.1 can be given the following sequential formulation: A function 1 E n* [a, b] if and only if there exists A E IRsuch that for any increasing sequence (en) in (a, b) with c -+ b, then I E n*[a, c ] and j,cn1 -+ A.

(a) Apply Hake'sTheorem to conclude that g(x) := l/x2/3forx E (O,I]andg(O) :=Obelongs to n*[O, 1]. (b) Explain why Hake's Theorem does not apply to I(x) := I/x for x E (0. 1] and 1(0) := 0 (which does not belong to n*[O, 1]).

Apply Hake's Theorem to g(x) := (1 - x)-1/2 for x E [0. 1) and g(1) := O.

Suppose that 1 E n*[a,c]forallc E (a, b) andthatthereexistsy E (a,b)andw E [y,b] such that I/(x)1 ::: w(x) for x E [y, b]. Show that 1 E n*[a, b].

Show that the function gl(x) :=x-I/2sin(1/x) for x E (O,I]andgl(O) :=Obelongsto[O, 1]. (This function was also considered in Exercise 10.1.15.)

Show that the following functions (properly defined when necessary) are in [0, 1]. x Inx b sin7rx (a) _1 2' () _I ' +x nx Inx (d) r;--'j' v 1 - x2

Determine whether the following integrals are convergentor divergent. (Define the integrands to be 0 where they are not already defined.) (a) o X3/2 ' 1 1 Inxdx (c) , o xJ 1 - x2 (e) 11 (lnx)(sin(1/x dx, (b) o X3/2 ' 1 1 lnx dx (d) -, o I-x 1 1 dx (t) 0 JX(1 - x>,

If 1 E n[a, b], showthat1 E [a, b].

If 1 E [a, b], showthat 12 is notnecessarilyin [a, b].

If I, g E [a, b] and if g is bounded and monotone, show that Ig E [a, b]. More exactly, if Ig(x)1 ~ B, show that II/gll ~ BII/II.

(a) Give an example of a function I E R*[O, 1] such that max{f,O} does not belong to R*[O, 1]. (b) Can you give an example of I E [0, 1] such that max{f, O}~ [0, I]?

Write out the details of the proofthat min{f, g} E R*[a, b] in Theorem 10.2.8 when ex~ I and ex~ g.

Write out the details of the proofs of Theorem 10.2.11.

Givean I E [a, b] with I not identically0, but suchthat 11/11 =O.

If I, g E [a, b], show that 111/11 - IIglll~ 1I/:l: gll.

Establish the easy part of the Completeness Theorem 10.2.12.

If In(x) := xn for n E N, show that In E [0, 1] and that IIIn II ~ O. Thus II/n - 011 ~ 0, where 0 denotes the function identically equal to O.

Let gn(x) := -1 for x E [-1, -I/n), let gn(x) := nx for x E [-I/n, I/n] and let gn(x) := 1 for x E (1In, 1]. Show that IIgm - gn II ~ 0 as m, n ~ 00, so that the Completeness Theorem 10.2.12 implies that there exists g E [-1, 1] such that (gn) converges to gin [-1,1]. Find such a function g.

Lethn(x) :=nforx E (0, I/n)andhn(x) :=Oelsewherein[O, 1]. Does there exist h E [0,1] such that IIhn - hll ~ O?

Let kn(x) := n for x E (0, II n2) and kn(x) := 0 elsewhere in [0, 1]. Does there exist k E [0, 1] such that IIkn- kll~ O?