#3 A Supremely Infimum Problem (Zorn 1.9 #8) Let S R be non-empty and bounded below. Let-S f-xlxES). Show that sup(-S) exists. Then show that -inf (S) sup(-S). This problem shows that the completenes...
(6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S. (6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S.
2. Let A be a non-empty subset of R bounded below. Show that inf (A) is a border point of A
Let A be a non empty subset of R that is bounded below and let a=inf A. If a a&A, prove thal x is a limit point of A
Problem 4 [20] Suppose A C R is a non-empty set, and sup(A), inf (A) exist. Show that sup(A) inf (A) if and only if |A. Problem 5 [25]
5. Let S be a non empty bounded subset of R. If a > 0, show that sup (as) = a sup S where as = {as : ES}. Let c = sup S, show ac = sup (aS). This is done by showing (a) ac is an upper bound of aS. (b) If y is another upper bound of as then ac S7 Both are done using definitions and the fact that c=sup S.
5. Let S be a non-empty bounded subset of R. If a > 0, show that sup (aS) = a sup S where aS = {as : s E S}. Let c = sup S, show ac = sup (aS). This is done by showing: (a) ac is an upper bound of aS. (b) If y is another upper bound of aS then ac < 7. Both are done using definitions and the fact that c=sup S.
Problem 3. (1) Let H be a Hilbert space and S, TE B(HH). Then, prove that ||ST|| ||||||||| (2) Let X, Y be Hilbert spaces and Te B(X,Y). Then, prove that ||1||| sup ||T3|1 TEX=1 Let X, Y be Banach spaces. Definition (review) We denote by B(X, Y) a set of all bounded linear operators T:X + Y with D(T) = X. B(X, Y) is a vector space. Definition (review) A linear operator T:X + Y is said to be...