From Wiki
A locally convex topological vector space is a topological vector space in which the origin has a local base of absolutely convex *absorbent* sets.
Also from Wiki
Locally convex topological vector spaces: here each point has a local base consisting of convex sets.
From Planetmath
Let $V$ be a topological vector space over a subfield of the complex numbers (usually taken to be $\mathbb{R}$ or $\mathbb{C}$ ). If the topology of $V$ has a basis where each member is a convex set, then $V$ is a locally convex topological vector space.
I was wondering if the three definitions are equivalent? Specifically,
- The last two definitions seem to agree with each other, because the union of local bases, each for each point, is a base of the topology, and the restriction of a base to a point is a local base of that point?
Between the first two definitions:
- Because any translation of any open subset is still open, so I think it doesn't matter to specified for a local base of the origin or a local base of every point?
- But I am not sure why the first definition requires "absolutely convex" and "absorbent" while the second just "convex"?
Thanks and regards!