Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(#12) Show 0 € int(S) implies Sº is bounded.

(#12) Show 0 € int(S) implies Sº is bounded.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
100%

This is from an optimization and convex analysis class

**Problem #12: Boundedness of the Polar Set** Show that if \( 0 \in \text{int}(S) \) implies that \( S^\circ \) is bounded. In this problem, we are asked to demonstrate that if the origin \( 0 \) is an interior point of the set \( S \), then the polar set \( S^\circ \) is bounded. This involves concepts from convex analysis and functional analysis, particularly concerning interior points and polar sets. Here’s an outline of the principles involved: 1. **Interior Points**: The notation \( 0 \in \text{int}(S) \) signifies that there exists some neighborhood around the origin that is completely contained within the set \( S \). This property is essential for showing the boundedness of the polar set. 2. **Polar Set**: The polar of a set \( S \), denoted \( S^\circ \), is defined as the collection of all vectors \( y \) such that the inner product \( \langle x, y \rangle \leq 1 \) for all \( x \) in \( S \). The concept of a polar set is important in duality theories and optimization. 3. **Boundedness**: A set is bounded if there exists some positive number \( M \) such that the norm of every element in the set is less than or equal to \( M \). Here, we show that \( S^\circ \) is bounded given the interior point condition. To prove this, you would typically leverage the separation theorems and the properties of convex sets to demonstrate that if the origin is an interior point of \( S \), then there is a radius within which the properties of \( S \) can be exploited to show the boundedness of \( S^\circ \). A detailed step-by-step proof would be covered in a lecture or textbook on advanced convex analysis.
Transcribed Image Text:**Problem #12: Boundedness of the Polar Set** Show that if \( 0 \in \text{int}(S) \) implies that \( S^\circ \) is bounded. In this problem, we are asked to demonstrate that if the origin \( 0 \) is an interior point of the set \( S \), then the polar set \( S^\circ \) is bounded. This involves concepts from convex analysis and functional analysis, particularly concerning interior points and polar sets. Here’s an outline of the principles involved: 1. **Interior Points**: The notation \( 0 \in \text{int}(S) \) signifies that there exists some neighborhood around the origin that is completely contained within the set \( S \). This property is essential for showing the boundedness of the polar set. 2. **Polar Set**: The polar of a set \( S \), denoted \( S^\circ \), is defined as the collection of all vectors \( y \) such that the inner product \( \langle x, y \rangle \leq 1 \) for all \( x \) in \( S \). The concept of a polar set is important in duality theories and optimization. 3. **Boundedness**: A set is bounded if there exists some positive number \( M \) such that the norm of every element in the set is less than or equal to \( M \). Here, we show that \( S^\circ \) is bounded given the interior point condition. To prove this, you would typically leverage the separation theorems and the properties of convex sets to demonstrate that if the origin is an interior point of \( S \), then there is a radius within which the properties of \( S \) can be exploited to show the boundedness of \( S^\circ \). A detailed step-by-step proof would be covered in a lecture or textbook on advanced convex analysis.
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS