Topological properties of a normed space are widely different depending on 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 be a normed space and be a proper closed linear subspace. For every , there exists such that and .
Proof. Let ; without loss of generality, we may suppose . Let and be such that
Let . Then for all ,
because , hence .
Theorem 1: Let be a normed space. The unit closed ball is compact if and only if is finite-dimensional.
Proof. Let . Because any finite-dimensional linear subspace of is closed, we can use lemma 1 to construct by induction a sequence satisfying
for all ; the sequence has infinitely many terms because is infinite-dimensional.
In particular when . Therefore, is a sequence in the sphere without converging subsequence: cannot be compact.
Corollary: Let be an infinite-dimensional normed space. Then the interior of any compact subspace is empty.
Theorem 2: Let be an infinite-dimensional normed space and be a compact subspace. Then is path connected.
Proof. Without loss of generality we may suppose that since is a homeomorphism for all . Because the sphere is path-connected, it is sufficient to show that any point can be connected to . If the statement is clear, so from now on suppose that .
As in the proof of theorem 1, there exists a sequence in satisfying for all .
Suppose by contradiction that every line meets , that is for every there exists such that .
Because is compact and , there exists (independent on ) such that . Therefore, there exists a subsequence converging to some .
Again by compactness, the sequence has a subsequence converging to some .
Therefore, the sequence converges to , a contradiction with when .
Theorem 3: For all , the -dimensional sphere is simply connected but not contractible.
According to the following lemma, our theorem is just a consequence of Brouwer’s fixed point theorem, proved in a previous post about topological degree theory:
Lemma 2: Let be a normed space. The following statements are equivalent:
- The unit sphere is not contractible,
- Any continuous map has a fixed point,
- is not a retract of the unit ball .
Proof. Suppose that there exists a continuous map without fixed point. Then
defines a homotopy between and , so is contractible. We proved
Suppose that there exists a retraction . Then has no fixed point. Indeed, if were a fixed point, we would have hence whereas by assumption; therefore, a contradiction with . We proved .
Suppose that there exists a homotopy between and a constant map . Then
defines a retraction of onto . We proved .
In order to opposite our theorem with a statement in infinite dimension, we want to introduce an infinite-dimensional sphere . A possible construction is the following:
Defintion: Let be the direct sum of countably many copies of endowed with the norm . Then we define .
Theorem 4: The infinite-dimensional sphere is contractible.
Proof. First, with
the map defines a homotopy between the identity and the shift operator . Then, with
the map defines a homotopy between and the constant map .
Therefore, is homotopic to , ie. is contractible.
To understand theorem 4, we may notice that where is included into thanks to ; in particular, is a subcomplex of . Then, because attaching a -cell to a CW complex does not change the -th homotopy group when , we deduce that
that is is weakly contractible. However, according to Whitehead theorem, a weakly contractible CW complex is in fact contractible, so is contractible.
Therefore, intuitively, we add cells to to kill successively the homotopy groups in order to make weakly contractible and finally contractible.