Some surprising but elementary facts in infinite-dimensional topology

Mathematical Notes

Topological properties of a normed space $latex X$ are widely different depending on $latex X$ is either finite-dimensional or infinite-dimensional; in fact, whole books are devoted to topology in infinite dimensions. Here, we propose some surprising but elementary facts in infinite-dimensional topology.

Lemma 1: Let $latex X$ be a normed space and $latex A subset X$ be a proper closed linear subspace. For every $latex epsilon>0$, there exists $latex x in X$ such that $latex |x|=1$ and $latex d(x,A) geq 1- epsilon$.

Proof. Let $latex epsilon>0$; without loss of generality, we may suppose $latex epsilon<1$. Let $latex y in X backslash A$ and $latex a_0 in A$ be such that

$latex displaystyle d(y,A) leq | y-a_0 | leq frac{d(y,A)}{1- epsilon}$.

Let $latex displaystyle x = frac{y-a_0}{|y-a_0|}$. Then for all $latex a in A$,

$latex displaystyle |x-a|= frac{y- (a_0+ |y-a_0| cdot a)}{|y-a_0|} geq frac{d(y,A)}{|y-a_0|} geq 1- epsilon$,

because $latex a_0+ |y-a_0|…

View original post 1,005 more words

Advertisements

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s