Metrizable topological vector space

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric).

Version

v1 · 2026-09-28 · History

Domain-specific #

10706

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Functional Analysis, Topological Vector Spaces → Mathematics

Core Idea

Metrizable topological vector space is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

Scope of Application

• Additive sequences. 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.

• Additive sequences. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions.

• Additive sequences. Because f is a nonnegative subadditive function satisfying f(0) = 0, as described in the article on sublinear functionals, f is uniformly continuous on X if and only if f is.

• Examples of paranorms. The function p(x) := |\sin (\pi x)| + \min { 2, |x| } is a paranorm on \R that is balanced but nevertheless equivalent to the usual norm on R.

• Examples of F-seminorms. A non-negative real-valued function on X is a seminorm if and only if it is a convex F-seminorm, or equivalently, if and only if it is a convex balanced G-seminorm.

Clarity

A clear use of Metrizable topological vector space names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric).

Manages Complexity

Metrizable topological vector space compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—it will now be shown by induction on k that if n{\bull} = \left(n1, \ldots, nk\right) consists of non-negative integers such that \sum 2^{- n{\bull}} \leq 2^{- M} for some integer M \geq 0 then \sum U{n{\bull}} \subseteq UM.—and the practical consequence—a topological.

Abstract Reasoning

1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
2. State the relation. Use the source-grounded identity: In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric).
3. Check operation and conditions. A pseudometric space is a pair (X, d) consisting of a set X and a pseudometric d on X such that X 's topology is identical to the topology on X induced by d.

Knowledge Transfer

Within the home domain. Knowledge about Metrizable topological vector space transfers literally when a new case preserves the same carrier type, relation, and recognition test. 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. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued.

Neighborhood in Abstraction Space

Metrizable topological vector space sits in a crowded region of the domain-specific corpus (35^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Topological & Functional-Analytic Spaces (13 abstractions)

Nearest neighbors

• Topological Algebra — 0.90
• T0 space — 0.89
• Hereditarily normal space — 0.88
• T4 Space — 0.88
• Countably quasi-barrelled space — 0.87

Computed from structural-signature embeddings · 2026-10-08