n] is a sequence of closed, absolutely convex neighborhoods of zero so that U = [[intersection].

n]) of absolutely convex neighborhoods of zero in (E, [[tau].

There exists a non-complete real normed space whose totally antiproximinal

absolutely convex subsets are not rare.

Since the closure B = cl(S(a)) is a closed, idempotent, bounded and absolutely convex subset in A (see, for example, [27], pp.

6) The absolutely convex hull of S [subset] A is the set

If B [subset or equal to] E is closed

absolutely convex bounded, then [E.

An

absolutely convex subset of a locally convex algebra (E, [tau]) is said to be bornivorous (resp.

Otherwise there is an

absolutely convex neighborhood of the origin U in (E',[sigma] (E', E)) and a strictly increasing sequence {[n.

An

absolutely convex subset of (E, [tau]) is said to be bornivorous (resp.

If every

absolutely convex, closed and bounded subset of the vector subspace F of E is locally compact, then F endowed with the [rho](E, E')-topology is a (LB)-space, where [rho](E, E') designates the topology on E of the uniform convergence on the

absolutely convex compact subsets of E'.

Let B be a closed

absolutely convex absorbing subset of [E.

0](E) and B(E) stand for the families of all

absolutely convex 0-neighborhoods and

absolutely convex bounded sets in E, respectively.