"The two notions of boundedness coincide for locally convex spaces" From Wiki

The boundedness condition for linear operators on normed spaces can be
  restated. An operator is bounded if it takes every bounded set to a
  bounded set, and here is meant the more general condition of
  boundedness for sets in a topological vector space (TVS): a set is
  bounded if and only if it is absorbed by every neighborhood of 0. Note
  that the two notions of boundedness coincide for locally convex
  spaces.

I was wondering what "the two notions of boundedness" are referred to? In other words, their definitions?
Are they for operators between TVSes, or for subsets of TVS?
Thanks and regards!
 A: As far as I can tell, the citation you give from Wikipedia is ambiguous at best,
and mistaken under one natural interpretation.
Here is a brief explanation of the relationship between continuity and boundedness for linear maps between topological vector spaces:
For a linear map $L$ between normed vector spaces spaces $V$ and $W$, there are (at least) two natural conditions you could impose: that $L$ is continuous, in the usual topological sense, or that $L$ is bounded, i.e. takes bounded subsets in $V$ to bounded subsets in $W$.   It turns out that these two conditions coincide (this is an easy exercise, using the definition of the metric in terms of the norm, and the linearity of $L$).
On an arbitrary topological vector space, one can also define the notion of bounded subsets, even though the topology is not defined in terms of a norm, or even a metric, in general.  The definition is the one given in your Wikipedia citation: a subset $B$ is bounded if given any neighbourhood $U$ of the origin, there is a scalar $\lambda$ such that $B \subset \lambda U$. (This is often phrased as in the paragraph you cite: $B$ is absorbed by $U$.)
In a normed space, it is easy to check that this notion coincides with the notion of bounded subsets as defined in terms of the norm.  
Given a linear map between topological vector spaces, we call it bounded if it takes bounded sets in the domain to boudned sets in the codomain.  One easily checks that continuous linear maps are necessarily bounded.  But it's not true in general that a bounded linear map between topological vector spaces is continuous. 
A locally convex topological vector space $V$ is called bornological if any bounded linear map from $V$ to another locally convex topological vector space $W$ is necessarily continuous.  The above discussion shows that normed spaces are bornological. (But as was already indicated, not all locally convex topological vector spaces are bornological, although most common ones are; e.g. Frechet spaces are, and locally convex inductive limits of bornological spaces are again bornological.)  
A: They are for subsets of normed linear spaces.


*

*In a normed linear space a set $X$ is bounded, iff there is some $C>0$ with $||x|| < C$ for all $x \in X$.

*Equivalently, a set is bounded iff it is absorbed by every neighbourhood of 0.


So a linear operator on normed spaces is bounded, iff it maps bounded sets to bounded sets.
This motivates the definition for more general topological vector spaces: An operator is called bounded, iff it maps bounded sets to bounded sets.
Such an operator does not need to be continuous, in contrast to normed spaces. Topological vector spaces where all bounded operators are continuous are called bornological spaces.
