Weakly l.s.c. function attains its min on weakly compact set?

• quasar987
In summary, the author claims that a weakly lower semi-continuous function attains its minimum on a convex weakly compact subset of a normed space without providing justification. However, the infimum of the function on the subset can be found and a sequence can be constructed such that the function converges to the infimum. This is possible because the subset is weakly compact and the function is weakly lower semi-continuous. The reason for believing that the subset is weakly sequentially compact can be understood through the Eberlein-Smulian theorem, although there may be a simpler explanation.

Homework Helper
Gold Member

Homework Statement

I'm reading the proof of a theorem and the author claims w/o justification that a weakly lower semi-continuous function (w.l.s.c.) f:C-->R attains its min on the convex weakly compact subset C of a normed space E.

At first I though I saw why: Let a be the inf of f on C and x_n be a sequence in C such that f(x_n) --> a. Since C is weakly compact, we can find a weakly convergent subsequence x_n_k-->x, and because f is w.l.s.c., we will have f(x)<=a, thus f(x)=a.

But what reason do we have to believe that C is weakly sequentially compact, so that the bold part above is justified??

(By "weakly" I mean "under the weak topology $$\sigma(C,C^*)$$".)

Last edited:
That a weakly compact set C is weakly sequentially compact is true, and follows from the Eberlein-Smulian theorem. But the Eberlein-Smulian is highly nontrivial, so there's probably an easier way to see why what you said is true.

1. What is a weakly l.s.c. function?

A weakly l.s.c. (lower semi-continuous) function is a mathematical function that satisfies the property that the limit of the function at any point is less than or equal to the value of the function at that point. In other words, the function does not have any sudden jumps or discontinuities.

2. What does it mean for a function to attain its min on a weakly compact set?

A function attains its min on a weakly compact set if there exists a point in the set where the function has its minimum value. In other words, the function reaches its lowest point within the weakly compact set.

3. How is the concept of weakly l.s.c. function related to optimization problems?

Weakly l.s.c. functions are commonly used in optimization problems because they ensure that a minimum value exists and can be found within a given set. This allows for efficient and accurate solutions to optimization problems.

4. Can a weakly l.s.c. function attain its min on a set that is not weakly compact?

Yes, a weakly l.s.c. function can attain its min on a set that is not weakly compact. However, in order for the function to have a minimum value, the set must have certain properties, such as being closed and bounded.

5. How is the concept of weakly l.s.c. function useful in real-world applications?

Weakly l.s.c. functions are useful in various real-world applications, such as economics, engineering, and physics. They can be used to model and optimize various systems, such as production processes, transportation networks, and energy consumption, to name a few.