Dimensionality Reducing by Alpha-Dense Curves ... - RUA

Observe that for any normed space
and, from the preceding theorem, in a Banach space, if and only if the space has
finite dimension.Then if the space has infinite dimension it follows that . Now, a
natural question is:
does it exist a normed space for each value between 0 and 1?
The answer is given by the following result (Mora and Mira, 2003).
Theorem
For any normed space, the degree of densificability of its unit ball can take exactly two
values, either 0 or 1.
Now, we obtain a well-known classical theorem as corollary the Riesz theorem on the
dimension:
Corollary
Any normed space locally compact is finite dimensional.