Locally Convex Topological Vector Space - Further Definitions and Properties

Further Definitions and Properties

• A family of seminorms {p_α}_α is called total or separated or is said to separate points if whenever p_α(x) = 0 holds for every α then x is necessarily 0. A locally convex space is Hausdorff if and only if it has a separated family of seminorms. Many authors take the Hausdorff criterion in the definition.
• A pseudometric is a generalisation of a metric which does not satisfy the condition that d(x,y) = 0 only when x = y. A locally convex space is pseudometrisable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is then given by
  (where the 1/2n can be replaced by any positive summable sequence a_n). This pseudometric is translation-invariant, but not homogeneous, meaning d(kx,ky) does not equal |k|d(x,y), and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and only if the family of seminorms is separated, since this is the case if and only if the space is Hausdorff. If furthermore the space is complete, the space is called a Fréchet space.
• As with any topological vector space, a locally convex space is also a uniform space. Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
• A Cauchy net in a locally convex space is a net {x_κ}_κ such that for every ε > 0 and every seminorm p_α, there exists a κ such that for every λ, μ > κ, p_α(x_λ–x_μ) < ε. In other words, the net must be Cauchy in all the seminorms simultaneously. The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrisable, general spaces may be defined by an uncountable family of pseudometrics. Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces. A locally convex space is complete if and only if every Cauchy net converges.
• A family of seminorms becomes a preordered set under the relation p_α ≤ p_β if and only if there exists an M > 0 such that for all x, p_α(x) ≤ Mp_β(x). One says it is a directed family of seminorms if the family is a directed set with addition as the join, in other words if for every α and β, there is a γ such that p_α + p_β ≤ p_γ. Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family {p_α}_α∈I, let Φ be the set of finite subsets of I, then for every F in Φ, define q_F=∑_α∈F p_α. One may check that {q_F}_F∈Φ is an equivalent directed family.
• If the topology of the space is induced from a single seminorm, then the space is seminormable. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if 0 has a bounded neighborhood.