Space of all sequences converging

Assignment Help Basic Statistics

Reference no: EM13990844

1. In any infinite-dimensional F -space, show that any compact set has empty interior.

2. A continuous linear operator T from one Banach space X into another one Y is called compact iff T {x ∈ X : lx l≤ 1} has compact closure. Show that if X and Y are infinite-dimensional, then T cannot be onto Y . Hint: Apply Problem 10.

3. Show that there is a separable Banach space X and a compact linear operator T from X into itself such that T {x : lx l ≤ 1} is not compact. Hint: Let X = c0, the space of all sequences converging to 0, with norm l{xn }l := supn |xn |.

Reference no: EM13990844

Questions Cloud

Baire and borel algebras : Show that there is a (non-Hausdorff) topology on a set X of two points for which the Baire and Borel σ-algebras are different.

Countable union of nowhere dense sets : Let µ be a finite Borel measure on a separable metric space S. As- sume µ({x }) = 0 for all x ∈ S. Prove that there is a set A of first cat- egory (a countable union of nowhere dense sets; see Theorem 2.5.2) with µ(S\ A) = 0.

Find the total inertia of the pendulum : Complex pendulum A pendulum is made of the following: A 2.4 meter long metal rod with a radius of 0.02 m and a mass of 24 kg. A circular weight of 12 kg attached to the rod at a distance of 1.8 m from the top. A square weight of 8 kg attached at t..

What did lbj company show as total debits : What did LBJ Company show as total debits? Which inventory method will result in the lowest income taxes when prices are decreasing?

Space of all sequences converging : Show that there is a separable Banach space X and a compact linear operator T from X into itself such that T {x : lx l ≤ 1} is not compact. Hint: Let X = c0, the space of all sequences converging to 0, with norm l{xn }l := supn |xn |.

Critical essay for philosophy : Critical Essay for philosophy-Choose an issue in professional ethics that pertains to a particular type of professional. Engineering is the best choice.

Sequence of bounded linear operators : Let (X, I·I) and (Y, |·|) be Banach spaces. Let Tn be a sequence of bounded linear operators from X into Y such that for all x in X, Tn (x ) converges to some T (x ). Show that T is a bounded linear operator.

What is the frequency of the most intense radiation emitted : What is the frequency of the most intense radiation emitted by your body? Assume a skin temperature of 95 Degree F. What is the wavelength of this radiation?

What is the epidemiological concern : What public health recommendation can you make to alleviate your identified epidemiological concern?

User Account

All Pages