Question: Let A and B be convex subsets in a finite-dimensional normed linear space X. A and B can be strongly separated if and only if

Let A and B be convex subsets in a finite-dimensional normed linear space X. A and B can be strongly separated if and only if
P(A, B) = inf{||x - y|| : x ∈ A, y ∈ B} > 0
Combining proposition 3.14 with corollary 3.2.3 gives the following important result, a geometric form of the Hahn-Banach theorem (proposition 3.15).

Step by Step Solution

★★★★★

3.34 Rating (163 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Assume inf x y x y 2 0Let 0 be the open ball around ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

914-M-N-A-O (530).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

Let SX be a closed and bounded subset of a finite-dimensional normed linear space X with basis {x1, x2, . . . ; xn}, and let xm be a sequence in S. Every term xm has a unique representation 1.Using...

Let A and B be nonempty, convex subsets in a normed linear space X with int A and int AB . Then A and B can be properly separated. Frequently stronger forms of separation are required. Two sets A and...

1. Let A and B be nonempty, disjoint, convex subsets in a normed linear space X. A and B can be strongly separated if and only if there exists a convex neighborhood of U of 0 such that (A + U) B = ...

Convex geometry 0.1 Dehn-Sommerville relations In this section we will talk about convex polytopes. One possible denition of a convex polytope uses the notion of a half-space: A closed half-space in...

Functional analysis Questions 81 17:36 - Expert Q&A W + 2. A subset M of a normed space X is said to be conver if x, y e M implies that { z EX : 2 = ax + ( 1 - c ) y, 0 s as1}cM, i.e. if the "line...

The Basics of Financial Mathematics Spring 2003 Richard F. Bass Department of Mathematics University of Connecticut c These notes are 2003 by Richard Bass. They may be used for personal use or class...

(a) In SystemVerilog, what is the difference between: (i) The ternary operator ? and if...then...else statements? [2 marks] (ii) always_ff and always_comb? [2 marks] (iii) Blocking, non-blocking and...

Let {S1, S2, . . . , Sn} be a collection of nonempty compact subsets of an m-dimensional linear space, and let x conv We consider the Cartesian product of the convex hulls of Si, namely Every point...

MATHEMATICS FOR MACHINE LEARNING Marc Peter Deisenroth A. Aldo Faisal Cheng Soon Ong Contents Foreword 1 Part I Mathematical Foundations 9 1 Introduction and Motivation 11 1.1 Finding Words for...

1. What is the issue being addressed in the paper? 2. What are the findings of the paper? 3. Why is this paper important to auditors, and what are the implications of this paper for the auditing...

Consider the following list of project activities: Duration Activity Predecessor (months) ABCDE A F G A DA C, D B, E F 5 6 3.5 4.5 6 5 3.5 a. Draw an Activity on Node (AON) diagram to reflect the...

You are a Chartered Accountant working for Apex Limited (Apex). You report to Tim Fairhall, the financial controller. Apex has entered into a lease agreement for a piece of equipment from National...

UTT, Inc., has a bond outstanding with a coupon rate of 6 . 2 7 percent and semiannual payments. The yield to maturity is 5 . 7 percent and the bond matures in 1 7 years. What is the market price if...

All of the following are generally part of the contract documents EXCEPT: Proposal Scope of work Terms and conditions Negotiation process

Consider the dissolved oxygen concentration at TVA dams first presented in Exercise 8-81. (a) Test the hypotheses H0: = 4 versus H1  4 Use a = 0.01. (b) What is the P-value of the test...

Consider the cigar tar content data first presented in Exercise 8-82. (a) Can you support a claim that mean tar content exceeds 1.5? Use a = 0.05 (b) What is the P-value of the test statistic...

Exercise 6-22 gave data on the heights of female engineering students at ASU. (a) Can you support a claim that mean height of female engineering students at ASU is 65 inches? Use a = 0.05 (b) What is...

q4 handwritting sol 4) (15 p.) The annual operating costs of Machine A are $2,000. The machine will perform satisfactorily over the next five years and has an estimated market value (MV) of $3,000 at...

Question 14 Calculate the values of Z, where Z ~ N(mean=0, variance=1), for which 2% of the data lies in the lower tail and 2% lies in the upper tail. Not yet answered Marked out of (answer in...

8. A company's current assets include cash = $100,000; accounts receivable = $100,000; and inventories = $200,000. Their gross fixed assets = $1 million and are depreciated by $100,000. While their...