In a proper infinite dimensional subspace of a infinite dimensional space, are all points boundary points?

Asked Oct 30 '22 at 08:31

Active Oct 30 '22 at 08:31

Viewed 57 times

In a proper infinite-dimensional subspace of an infinite-dimensional space, are all points boundary points? (Assume we are working with a normed Banach space).

So this is intuitively true for finite-dimensional space like $R^3$ where all planes and lines do not contain any open sets, so in fact, every point is also a boundary point. But is this true in the infinite-dimensional case where the subspace is also infinite-dimensional?

asked Oct 30 '22 at 08:31

Bill

4,417

2

No proper subspace of a normed linear space has any interior points, so all points are boundary points. – geetha290krm Oct 30 '22 at 08:34
Does this answer your question? Every proper subspace of a normed vector space has empty interior – Anne Bauval Oct 30 '22 at 09:10

In a proper infinite dimensional subspace of a infinite dimensional space, are all points boundary points?

0 Answers0