Basic Analysis – Gently Done Topological Vector Spaces

58 Basic Analysis
Proposition 6.13 Any neighbourhood of 0 in a topological vector space X is
absorbing.
Proof For any x ∈ X and neighbourhood V of 0, the continuity of scalar multiplication
(at (0,x)) ensures the existence of δ > 0 such that tx ∈ V whenever
|t| < δ.
In the vector space R 3 , we can readily visualise the proper subspaces (lines or
planes) and we see that these can have no interior points, since any open set must
contain a ball. In particular, no such subspace can be open. This result is quite
general as we now show.
Proposition 6.14 The only open linear subspace in a topological vector space X
is X itself.
Proof Suppose that M is an open linear subspace in a topological vector space X.
Then 0 belongs to M so there is a neighbourhood V of 0 in X such that V ⊆ M.
Let x ∈ X. Since V is absorbing, there is δ > 0 such that tx ∈ V whenever |t| < δ.
In particular, tx ∈ M for some (suitably small) t with t �= 0. But M is linear, so
it follows that x = t −1 tx ∈ M, and we have M = X.
In fact, a proper linear subspace M of a topological vector space X can have
no interior points at all. To see this, suppose that x is an interior point of M.
Then there is a neighbourhood U of x with U ⊆ M. But then −x + U belongs
to M and is a neighbourhood of zero. We now argue as before to conclude that
M = X.
Definition 6.15 A subset non-empty subset B in a vector space X is said to be
balanced if tB ⊆ B for all t ∈ K with |t| ≤ 1.
Note that any balanced set contains 0 and that if B is balanced, then B ⊆ rB
for all r with |r| ≥ 1.
Example 6.16 The balanced subsets of C, considered as a complex vector space,
are all the discs (either with or without their boundary) withcentre at 0—together
with C itself. If C is considered as a vector space over R, then the balanced sets
are given by unions of line segments symmetrical about and passing through 0.
For example, the rectangle {z ∈ C : |Rez| < 1 and |Imz| ≤ 1} is a balanced set
in C when treated as a real vector space.
Department of Mathematics King’s College, London