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1. The problem statement, all variables and given/known data

Let C be a closed convex set in a Banach space with non empty interior. Show that cl(int(C))=C (show that the closure of the interior of C is S itself).

3. The attempt at a solution

Since int(C) is included in C, we have cl(int(C)) [itex]\subset[/itex] cl(C)=C.

For the other inclusion, let x be in C. I'm going to try to find a sequence in int(C) that converges to x. This will prove that C [itex]\subset[/itex] cl(int(C)).

Let y be in int(C) and B(y;r) be a ball centered on y and entirely contained in int(C). Since C is convex, the line d joining y and x is entirely contained in C. If I can show that d\{x} is entirely contained in int(C), then I will have won.

My idea is that not only is d contained in C, but so is the line from any point in B(y;r) to x! The set of all these lines make up a set the form of an ice cream cone where x is the tip and the ball B(y;r) is the ice cream ball.

So, I can either show that this ice cream cone minus {x} is open, or find explicitly for each point of d\{x} an open ball entirely contained in the ice cream cone.

In either case, how do we do that?!?

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# Homework Help: Show set is open - technical difficulties

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