Confusing notions of closed

[Poirot]

I'm working in a normed space X. A subset E of X is closed if it's complement in X is open.
I've also read that in an inner product space, a set E is closed if all convergent sequences in E converge to something in E.

My question is, for a normed space X, are these 2 definitions equivalent?

 

[Sudharaka]

I'm working in a normed space X. A subset E of X is closed if it's complement in X is open.
I've also read that in an inner product space, a set E is closed if all convergent sequences in E converge to something in E.

My question is, for a normed space X, are these 2 definitions equivalent?

Hi Poirot,

Yes they are. Refer Lemma 2.2.3 on the following link.

http://www.google.ca/url?sa=t&rct=j...DgUUKXmasG-n7lCX-1BYqDw&bvm=bv.53537100,d.dmg