In functional analysis and related areas of mathematics, a metrizable (resp.
An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
ranges over the positive real numbers, forms a basis for a topology on
This is because despite it making addition and negation continuous, a group topology on a vector space
with the following properties:[2] where we call a G-seminorm a G-norm if it satisfies the additional condition: If
This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms.
Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions.
These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable.
The following theorem is true more generally for commutative additive topological groups.
always denotes a finite sequence of non-negative integers and use the notation:
form basis of balanced neighborhoods of the origin then it may be shown that for any
ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set.
ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of open sets.
that also forms a neighborhood basis at the origin for a vector topology on
is increasing, a basis of open neighborhoods of the origin consists of all sets of the form
This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.
: The Fréchet combination can be generalized by use of a bounded remetrization function.
A bounded remetrization function[15] is a continuous non-negative non-decreasing map
is translation invariant and absolutely homogeneous, which means that for all scalars
[26] A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin.
Consequently, any metrizable TVS that is not normable must be infinite dimensional.
is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then
are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover,
then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.
[14] Theorem[29] — All infinite-dimensional separable complete metrizable TVS are homeomorphic.
is a pseudometrizable TVS whose topology is induced by a translation invariant pseudometric
is a closed vector subspace of a complete pseudometrizable TVS
[14] Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.
[22] The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.
has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.