Metrizable topological vector space

If X is a vector space over the real or complex numbers then an F-seminorm on X is a non-negative map p : X → [0, ∞) with the following properties:[1]

1. p(x+y) ≤ p(x) + p(y) for all x, y ∈ X;
2. p(ax) ≤ p(x) for all x ∈ X and all scalars a satisfying |a| ≤ 1;
3. ${\displaystyle \lim _{n\to \infty }p\left({\frac {x}{n}}\right)=0}$ for all x ∈ X.

An F-seminorm is called an F-norm if in addition p(x) = 0 implies x = 0.

If X is a TVS then the following are equivalent:[1]

1. X is pseudometrizable.
2. The topology of X is induced by an F-seminorm.
3. X has a countable neighborhood base at the origin.

If X is a TVS then the following are equivalent:[1]

1. X is metrizable.
2. X is Hausdorff and pseudometrizable.
3. The topology of X is induced by an F-norm.
4. X is Hausdorff and has a countable neighborhood base at the origin.

If X is a locally convex TVS then the following are equivalent:[2]

1. X is pseudometrizable.
2. X has a countable neighborhood base at the origin consisting of convex sets.
3. The topology of X is induced by a countable family of (continuous) seminorms.
4. The topology of X is induced by a countable increasing sequence of (continuous) seminorms ${\displaystyle \left(p_{i}\right)_{i=1}^{\infty }}$ (increasing means that for all i, ${\displaystyle p_{i}\leq p_{i+1}}$).

If the topology of X is induced by a countable increasing sequence of continuous seminorms ${\displaystyle \left(p_{i}\right)_{i=1}^{\infty }}$, then the topology of X is induced by the following F-seminorm on X:

${\displaystyle p(x):=\sum _{i=1}^{\infty }{\frac {p_{i}(x)}{2^{i}\left[1+p_{i}(x)\right]}}}$

where if X is Hausdorff then p is an F-norm.[2] Furthermore, assuming that ${\displaystyle \left(p_{i}\right)_{i=1}^{\infty }}$ is increasing, a basis of open neighborhoods at the origin consists of all sets of the form ${\displaystyle \left\{x\in X;p_{i}(x)<r\right\}}$ as i ranges over all positive integers and r > 0 ranges over all positive real numbers. The translation invariant pseudometric on X induced by this F-seminorm is the

${\displaystyle d(x,y)=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {p_{i}(x-y)}{1+p_{i}(x-y)}}}$

(this metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations).[3]

If a TVS X has a bounded neighborhood of 0 then it is pseudometrizable; the converse is in general false.[3] If a Hausdorff TVS X has a bounded neighborhood of 0 then it is ometrizable.[3] If a TVS X has a convex bounded neighborhood of 0 then it is seminormable; if in addition X is Hausdorff then it is normable.[3]

If X is a pseudometrizable TVS and M is a vector subspace of X then X/M is pseudometrizable; furthermore, if p is an F-seminorm on X inducing X's topology then the map ${\displaystyle {\overline {p}}:X/M\to \mathbb {R} }$ defined by ${\displaystyle {\overline {p}}\left(x+M\right):=\inf \left\{p(x+m):m\in M\right\}}$ is an F-seminorm on X/M that induces the quotient topology on X/M. If p is an F-norm on X that induces the topology on X and if M is a closed vector subspace of X then X/M is metrizable and its topology is induced by ${\displaystyle {\overline {p}}:X/M\to \mathbb {R} }$.[1]

If X is a Hausdorff locally convex TVS then the following are equivalent:

1. X is normable.
2. X has a bounded neighborhood of the origin.
3. the strong dual ${\displaystyle X_{b}^{\prime }}$ of X is normable.[4]
4. the strong dual ${\displaystyle X_{b}^{\prime }}$ of X is metrizable.[4]

Suppose that (X, d) is a pseudometric space and B ⊆ X. We say that B is metrically bounded or d-bounded if there exists a real number R > 0 such that d(x, y) ≤ R for all x, y ∈ B; the smallest such R is then called the diameter or d-diameter of B.[3] If B is bounded in the TVS X then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.[3]

Every metrizable locally convex TVS is a bornological space and a Mackey space.

If X is a complete pseudo-metrizable TVS and M is a closed vector subspace of X, then X/M is complete.[1]

If X is a metrizable locally convex space, then the strong dual of X is bornological if and only if it is infrabarreled, if and only if it is barreled.[5]

Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.[6] Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.[6]

In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.[7]

If X is a pseudometrizable TVS and A maps bounded subsets of X to bounded subsets of Y, then A is continuous.[3]

Theorem (Mackey's countability condition):[3] Suppose that X is a metrizable locally convex TVS and that ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ is a countable sequence of bounded subsets of X. Then there exists a bounded subset B of X and a sequence ${\displaystyle \left(s_{i}\right)_{i=1}^{\infty }}$ of positive real numbers such that B_i ⊆ s_i B for all i.

• Metric space

• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.CS1 maint: ref=harv (link)
• Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1.CS1 maint: ref=harv (link)
• Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link)
• Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 1-58488-866-0. OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 65–75.
• Schaefer, Helmut H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Treves, Francois (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link)