Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2025 • Question 1

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2025 • Question 1

Show that for the sphere \(S(0, 1)\) in a real normed space \(X\) and any point \(x_0 \in S(0, 1)\) there is a hyperplane \(H_0 \ni x_0\) such that the closed ball \(\widetilde {B}(0, 1)\) lies entirely in one of the two closed half spaces determined by \(H_0\).

Raw LaTeX

Show that for the sphere \(S(0, 1)\) in a real normed space \(X\) and any point \(x_0 \in S(0, 1)\)
there is a hyperplane \(H_0 \ni x_0\) such that the closed ball \(\widetilde{B}(0, 1)\) lies entirely in one of the
two closed half spaces determined by \(H_0\).

Topic Tags

Banach and Normed Spaces Separation Theorems and Supporting Hyperplanes