## ★ Bornivorous set

In functional analysis, a subset of a real or complex vector space X that has an associated vector bornology ℬ is called bornivorous and bornivorous if it absorbs each element ℬ. If X is a topological vector space, then a subset of S X is bornivorous if it is bornivorous with respect to the von Neumann bornology of X.

Bornoid sets play an important role in defining many classes of topological vector spaces, for example, Bornological spaces.

## 1. Definitions. * (Определения) *

If X is TVS and if A and B are subsets of X, then we say that A absorbs B if there exists a real number r & gt; 0 such that B ⊆ sA for all scalars s such that | s | ≥ r.

If X is TVS, then a subset of S of X is bornivorous if S absorbs every von Neumann-bounded subset of X. a disk in TVS is called infrabornivorous if it absorbs every Banach disk. In Hausdorff locally convex televisions, a disk is infra-bivalent if and only if it absorbs all CDs.

## 2. Examples and sufficient conditions. * (Примеры и достаточные условия) *

Every area of origin on TV is born a carnivore. The convex hull, closed convex hull, and balanced hull of the bornivorous set are again bornivorous.

### Counter examples. * (Встречные примеры) *

Let X - R 2 {\displaystyle \mathbb {R} ^{2}} be a vector space over reals. If is a balanced shell of a closed line segment between -1 and 1 then is not bornivorous, but the convex shell of is bornivorous. If T is a closed and "filled" triangle with vertices -1, -1, -1 and 1 then T is a convex set that is not bornivorous, but its balanced shell is bornivorous.

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