to e.g. Statement (3) may become: For every there exists a scalar such that If is a locally convex space whose topology is defined by a family of continuous seminorms, then this list may be extended to include: is bounded for all There exists a sequence of non-zero scalars such that for every sequence in the sequence is bounded in (according to any defining condition other than this one). For all is bounded (according to any defining condition other than this one) in the semi normed space If is a normed space with norm (or more generally, if it is a seminormed space and is merely a seminorm), then this list may be extended to include: is a norm bounded subset of By definition, this means that there exists a real number such that for all Thus, if is a linear map between two normed (or seminormed) spaces and if is the closed (alternatively, open) unit ball in centered at the origin, then is a bounded linear operator (which recall means that its operator norm is finite) if and only if the image of this ball under is a norm bounded subset of is a subset of some (open or closed) ball. This ball need not be centered at the origin, but its radius must (as usual) be positive and finite. If is a vector subspace of the TVS then this list may be extended to include: is contained in the closure of In other words, a vector subspace of is bounded if and only if it is a subset of (the vector space) Recall that is a Hausdorff space if and only if is closed in So the only bounded vector subspace of a Hausdorff TVS is A subset that is not bounded is called . Bornology and fundamental systems of bounded sets The collection of all bounded sets on a topological vector space is called the or the () A or of is a set of bounded subsets of such that every bounded subset of is a subset of some The set of all bounded subsets of trivially forms a fundamental system of bounded sets of Examples In any locally convex TVS, the set of closed and bounded disks are a base of bounded set. Examples and sufficient conditions Unless indicated otherwise, a topological vector space (TVS) need not be Hausdorff nor locally convex. Finite sets are bounded.