We consider the problem

min{f(x) : x V } (4.1)

where V is a normed vector space and f : V R.

Remark 4.1 This problem is technically very similar to the problem

min{f(x) : x } (4.2)

whereis an open set in V , as shown in the following exercise.

Exercise 4.1 Suppose V is open. Prove carefully, using the definitions given earlier,

that x is a local minimizer for (4.2) if and only if x is a local minimizer for (4.1) and x .

Further, prove that the "if" direction is true for general (not necessarily open) , and show

by exhibiting a simple counterexample that the "only if" direction is not.

We consider the problem min {f(x) : x EV} (4.1) where V is a normed vector space and f : V - R. Remark 4.1 This problem is technically very similar to the problem min{f(x) : xen} (4.2) where 2 is an open set in V, as shown in the following exercise. Exercise 4.1 Suppose & C V is open. Prove carefully, using the definitions given earlier, that x is a local minimizer for (4.2) if and only if x is a local minimizer for (4.1) and x En. Further, prove that the "if" direction is true for general (not necessarily open) 2, and show by exhibiting a simple counterexample that the "only if" direction is not