Vector Bornologies | 13 | Introductory Theory of Topological Vector SP

ABSTRACT

It is well-known that unit balls in normed spaces are neighbourhoods of 0 as well as bounded sets. In section 7, we have extended the notion of unit balls from the topological point of view. This section and next parallely deal with a generalization of the notion of unit balls from the boundedness, called vector bornologies. as defined by the following: 13.1 Definition.

Let X be a vector space. A family G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2016.tif"/> of subsets of X is called a vector bornology if it satisfies the following conditions:

(VB1) X = ∪ { B : B ∈ G } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2017.tif"/> .

(VB2) Any subset of any member in G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2018.tif"/> belongs to G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2019.tif"/> .

(VB3) B 1 + ⋯ + B n ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2020.tif"/> whenever B i ∈ G ( i = 1 , ⋯ ,n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2021.tif"/> .

(VB4) λB ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2022.tif"/> whenever B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2023.tif"/> and λ ∈ K.

(VB5) The circled hull of any B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2024.tif"/> belongs to G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2025.tif"/> .

Members in G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2026.tif"/> are called bounded sets.

A vector bornololgy G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2027.tif"/> on X is said to be :

(a) separated if G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2028.tif"/> does not contain any vector subspace of X different from {0};

(b) convex bornology if ΓB ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2029.tif"/> whenever B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2030.tif"/> .

A vector space equipped with a vector bornology (resp. convex bornology) is called a bornological vector space, abbreviated BVS (resp. convex bornological vector space, abbreviated CBVS). Hereafter we shall generally denote a BVS by (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2031.tif"/> ) or X G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2032.tif"/> or simply by X if the vector bornology G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2033.tif"/> does not require any special notation.

Any fundamental subfamily B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2034.tif"/> of a vector bornology G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2035.tif"/> on X is called a base for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2036.tif"/> (namely, any member in G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2037.tif"/> is contained in some element of B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2038.tif"/> ).

If G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2039.tif"/> is a vector bornology (resp. convex bornology) on X, then the family {circled hull of B : B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2040.tif"/> } (resp. { ΓB : B ∈ G } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2041.tif"/> is a base for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2042.tif"/> . Moreover, it is not hard to show that the following holds.

13.a <target id="page_181" target-type="page">181</target>Bases of vector bornologies:

Let X be a vector space.

(1) A family B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2043.tif"/> of subsets of X is a base for a vector bornology (resp. convex bornology) if and only if B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2044.tif"/> satisfies the following conditions:

(i) X = ∪ { B : B ∈ B } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2045.tif"/> ;.

(ii) for any finite B 1 ⋯ , B n ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2046.tif"/> , there is an B ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2047.tif"/> such that B 1 + ⋯ + B n ⊂ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2048.tif"/> ;

(iii) for any A ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2049.tif"/> and λ ∈ K, there is an B ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2050.tif"/> such that λA ⊂ B;

(iv) for any A ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2051.tif"/> the circled hull of A (resp. ΓA) is contained in some member in B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2052.tif"/> .

(2) Let B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2053.tif"/> be a base for a vector bornology (resp convex bornology) on X. Then the family, defined by M ( B ) = { D ⊂ X : D ⊂ B for some B ∈ B } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2054.tif"/>

is a vector bornology (resp. convex bornology) with B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2055.tif"/> as a base; M ( B ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2056.tif"/> is called the vector bornology (resp. convex bornology) generated by B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2057.tif"/> .

13.2 Definition.

Let G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2058.tif"/> ₁ and G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2059.tif"/> ₂ be two vector bornologies on X. We say that G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2060.tif"/> ₁ is coarser than G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2061.tif"/> ₂ (or G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2062.tif"/> ₂ is finer than G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2063.tif"/> ₁), denoted by G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2064.tif"/> ₁ ≤ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2065.tif"/> ₂, if G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2066.tif"/> ₂ ⊂ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2067.tif"/> ₁.

Remark

Suppose that B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2068.tif"/> ₁ and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2069.tif"/> ₂ are two bases for two vector bornologies on X, and that M ( B 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2070.tif"/> is the vector bornology generated by B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2071.tif"/> ₁ (for definition, see (13.a)(2)). If M ( B 1 ) ≤ M ( B 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2072.tif"/> , then we write B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2073.tif"/> ₁ ≤ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2074.tif"/> ₂ and also say that B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2075.tif"/> ₁ is coarser than B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2076.tif"/> ₂ (or B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2077.tif"/> ₂ is finer than B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2078.tif"/> ₁). It is clear that B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2079.tif"/> ₁ ≤ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2080.tif"/> ₂ if and only if for any D ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2081.tif"/> ₂ there is a B ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2082.tif"/> ₁ such that D ⊆ B.

13.3 Definition.

Let X G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2083.tif"/> and Y M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2084.tif"/> be BVS. A T ∈ L * ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2085.tif"/> is called a locally bounded map (w.r.t. G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2086.tif"/> and M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2087.tif"/> ) if T ( G ) = { T(B) : B ∈ B } ⊂ _ M . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2088.tif"/>

182The set consisting of all locally bounded maps (w.r.t. G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2089.tif"/> and M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2090.tif"/> ), denoted by L X ( X G , Y M ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2091.tif"/> (or simply L X ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2092.tif"/> ), is a vector subspace of L * ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2093.tif"/> .

A bijective linear map T : X → Y is called a bornological isomorphism (w.r.t. G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2094.tif"/> and M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2095.tif"/> if T and T^-1 are locally bounded (w.r.t. G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2096.tif"/> and M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2097.tif"/> ).

A linear functional f on X is said to be G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2098.tif"/> -bounded (or locally bounded for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2099.tif"/> ) if it is bounded on any B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2100.tif"/> . The set consisting of all G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2101.tif"/> -bounded linear functionals on X, denoted by X^x or ( X G ) x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2102.tif"/> , is a vector subspace of X*; X^x is called the bornological dual of X G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2103.tif"/> .

Remark

Let B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2104.tif"/> _X and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2105.tif"/> _Y be two bases for vector bornologies on X and Y resp. and T ∈ L * ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2106.tif"/> . Then T is locally bounded (w.r.t. B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2107.tif"/> _X and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2108.tif"/> _Y) if and only if for any D ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2109.tif"/> _X there is an B ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2110.tif"/> _Y such that T(D) ⊂ B.

13.4 Examples

(a) Let (X, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2111.tif"/> ) be a TVS.

(i) M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2112.tif"/> _von (the family of all P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2113.tif"/> -bounded subsets of X) is a vector bornology, called the von Neumann bornology by Hogbe–Nlend [1977, p.21] and denoted by M von X ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2114.tif"/> or M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2115.tif"/> (or M von X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2116.tif"/> or M von https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2117.tif"/> ); the family B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2118.tif"/> _von of all closed, circled P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2119.tif"/> -bounded subsets of X is a base for M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2120.tif"/> _von.

(ii) M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2121.tif"/> _p (the family of all P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2122.tif"/> -totally bounded subsets of X) is a vector bornology, called the precompact bornology and denoted by M p X ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2123.tif"/> or M p ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2124.tif"/> ( M p X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2125.tif"/> or M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2126.tif"/> _p) the family B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2127.tif"/> _p of all P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2128.tif"/> -closed circled, totally bounded subsets of X is a base for M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2129.tif"/> _p. Moreover, if P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2130.tif"/> is Hausdorff, then M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2131.tif"/> _p is the family of all precompact subsets of X.

(iii) M c ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2132.tif"/> (the family of all relatively P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2133.tif"/> -compact sets in X) is a vector bornology, called the compact bornology and denoted by M c X ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2134.tif"/> or M c ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2135.tif"/> (or M c X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2136.tif"/> or M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2137.tif"/> _c), the family B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2138.tif"/> _c of all compact circled sets is a base for M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2139.tif"/> _c.

(iv) The family B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2140.tif"/> _cd of all compact disks in X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2141.tif"/> is a base for a convex 183bornology M ( B cd ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2142.tif"/> which is called the convex compact bornology and denoted by M cd ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2143.tif"/> or M cd X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2144.tif"/> or M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2145.tif"/> _cd.

If P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2146.tif"/> is Hausdorff, then all M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2147.tif"/> _von, M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2148.tif"/> _p, M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2149.tif"/> _c and M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2150.tif"/> _cd are separated.

(b) Suppose that X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2151.tif"/> is a LCS.

(i) M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2152.tif"/> _von and M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2153.tif"/> _p are separated convex bornologies.

(ii) M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2154.tif"/> _c is, in general, not a convex bornology (since the convex hull of a P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2155.tif"/> -compact subset of X is, in general, not P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2156.tif"/> -compact). If, in addition, X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2157.tif"/> is quasi-complete then M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2158.tif"/> _c is a convex bornology.

(iii) The family, defined by B f = { Γ { x 1 , ⋯ , x n } : for any finite set { x 1 , ⋯ , x n } in X , n ≥ 1 } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2159.tif"/>

is a base for a convex bornology, which is called the finite-dimensional bornology and denoted by M f ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2160.tif"/> or M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2161.tif"/> _f (i.e. M f = M ( B f ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2162.tif"/> ).

(iv) Recall that a bounded disk B in X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2163.tif"/> is said to be infracomplete if ( X ( B ) , γ B ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2164.tif"/> is a Banach space. The family B inf ( P ) = { B = ΓB ⊂ X : B is ( bounded ) infracomplete } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2165.tif"/>

is a base for a convex bornology, which is called the infracomplete bornology (complete bornology in the terminology of Hogbe–Nlend [1977, p.42]), and denoted by M inf ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2166.tif"/> or M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2167.tif"/> _inf (i.e. M inf = M ( B inf ( P ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2168.tif"/> ).

(v) Let Y be a TVS. A subset H of L(X,Y) (the vector space of all continuous linear maps) is said to be equicontinuous if for any 0–neighbourhood U in Y, the set, defined by H − 1 ( U ) = ∩ T ∈ H T − 1 ( U ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2169.tif"/>

is a 0–neighbourhood in X. The family of all equicontinuous subsets of L(X,Y) is a vector 184bornology (convex bornology when Y is a LCS), which is called the equicontinuous bornology on L(X,Y) and denoted by M eq ( L ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2170.tif"/> .

(vi) On any LCS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2171.tif"/> , we have M f ⊂ M cd ⊂ M inf ⊂ M von ⊂ and M cd ⊂ M p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2172.tif"/>

or, equivalently, M von < M inf < M cd < M f < and M p < M cd . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2173.tif"/>

13.5 Definition.

Let G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2174.tif"/> be a vector bornology on X, let { x n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2175.tif"/> be a sequence in X and A ⊆ X.

(a) { x n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2176.tif"/> is called a b–null sequence (or Mackey–null sequence) for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2177.tif"/> if there is a circled B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2178.tif"/> and a sequence { λ n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2179.tif"/> in K such that lim n λ n = 0 and x n ∈ λ n B ( for all n ≥ 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2180.tif"/>

(b) If u ∈ X is such that { x n − u } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2181.tif"/> is a b–null sequence for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2182.tif"/> , then we say that { x n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2183.tif"/> is b–convergent (for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2184.tif"/> ) or b–convergent to u, denoted by X n → M u . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2185.tif"/>

(c) A is said to be b–closed (or Mackey–closed) (for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2186.tif"/> ) if the conditions x n ∈ A and x n → M u for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2187.tif"/>

imply that u ∈ A.

Remarks

(i) It is clear that if x n → M x,y n → M y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2188.tif"/> , and λ n → λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2189.tif"/> in K, then x n + y n → M x + y and λ n x n → M λx . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2190.tif"/>

185It is also trivial that every b–convergent sequence for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2191.tif"/> must be bounded, and that the image of a b–convergent sequence under a locally bounded map is again a b–convergent sequence.

(ii) The intersection of a family of b–closed subsets of (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2192.tif"/> ) is b–closed (for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2193.tif"/> ), hence the b–closure of A (for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2194.tif"/> ), denoted by A ˜ b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2195.tif"/> , is defined to be the intersection of all b–closed sets (for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2196.tif"/> ) containing A. Moreover, if A is a vector subspace, then so is A ˜ b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2197.tif"/> .

(iii) Let X and Y be BVS and T ∈ L * ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2198.tif"/> . If N is b–closed in Y, then T^-1(N) is b–closed in X.

13.b

For a vector bornology G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2199.tif"/> on X, the following statements are equivalent:

(a) G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2200.tif"/> is separated.

(b) Every b–convergent sequence in X (for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2201.tif"/> )has a unique limit.

(c) {0} is b–closed (for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2202.tif"/> ).

13.6 proposition

Let G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2203.tif"/> be a vector bornology on X and { x n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2204.tif"/> a sequence in X. Then the following statements are equivalent.

(a) { x n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2205.tif"/> is a b–null sequence (for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2206.tif"/> ).

(b) There exists a circled B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2207.tif"/> and a, decreasing sequence { ζ n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2208.tif"/> of positive numbers such that lim ζ n = 0 and _ x n ∈ ζ n B ( for all n ≥ 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2209.tif"/>

(c) There exists a circled B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2210.tif"/> such that for any ϵ > 0 one can find a natural number N(ϵ) with x n ∈ ϵ B for all _ n ≥ N ( ϵ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2211.tif"/>

If G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2212.tif"/> is a convex bornology. then one of (a),(b) and (c) is equivalent to the following:

186(d) There exists a disk B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2213.tif"/> such that x n ∈ X ( B ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2214.tif"/> and γ B ( x n ) → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2215.tif"/> .

Proof.

(a) ⇒ (b): Let B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2216.tif"/> be circled and let λ_n ∈ K be such that lim n λ n = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2217.tif"/> and x_n ∈ λ_nB (for all n ≥ 1). For any positive integer k, there is an N_k ≥ 1 such that | λ n | ≤ 1 k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2218.tif"/> (for all n ≥ N_k), hence λ n B ⊂ k -1 Β ( for all n ≥ N k ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2219.tif"/>

Without loss of generality one can assume that N_k is strictly increasing. For any j ≥ 1 with N_k ≤ j < N_k+1 (k ≥ 1), let ζ j = 1 k . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2220.tif"/>

Then the sequence { ζ j } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2221.tif"/>

has the required properties.

(b) ⇒ (c): Trivial.

(c) ⇒ (a): For any n ≥ 1, let ϵ n = inf { ϵ > 0 : x n ∈ ϵ B } and λ n = ϵ n + 1 n . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2222.tif"/>

Then λ_n → 0 and x_n ∈ λ_nB (for all n ≥ 1).

Finally, we assume that G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2223.tif"/> is a convex bornology, then the implication (d) ⇒ (a) holds with λ n = γ B ( x n ) + 1 n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2224.tif"/> (for all n ≥ 1). On the other hand, (b) implies x_n ∈ X(B) and γ B ( x n ) ≤ ζ n → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2225.tif"/> . Thus the implication (b) ⇒ (d) follows.

Remark

Because of (d), we say that a subset K of a separated CBVS (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2226.tif"/> ) is b–compact (resp. b–precompact) if there is a disk B ∈ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2227.tif"/> such that K is relatively compact (resp. totally bounded) in the normed space (X(B),γ_B).

13.7 Definition.

Let { ( X i , G i ) : i ∈ Λ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2228.tif"/> be a family of BVS (resp.CBVS) and π j : Π i ∈ Λ X i → X j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2229.tif"/> the j–th projection. The family Π i ∈ Λ G i = { A ⊂ _ Π i ∈ Λ X i : π i ( A ) ∈ G i ( for all i ∈ Λ ) } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2230.tif"/>

187defines obviously a vector bornology (resp. convex bornology) on πX_i, called the product bornology, and the pair ( Π X i , Π G i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2231.tif"/> is called the bornological product space.

Remarks

(i) The product bornology admits a base consisting of sets of the form { Π i ∈ Λ B i : B i ∈ G i ( i ∈ Λ ) } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2232.tif"/>

Moreover, Π i ∈ Λ G i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2233.tif"/> is the coarsest vector (resp. convex) bornology on Π i ∈ Λ X i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2234.tif"/> for which all projections π_i are locally bounded.

(ii) If each vector bornology G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2235.tif"/> ₁ on X is separated, then so is the product bornology Π i G i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2236.tif"/> .

Let (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2237.tif"/> ) be a BVS (resp. CBVS) and let M be a vector subspace of X. Then the family M ∩ G = { M ∩ B : B ∈ G } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2238.tif"/>

is a base for a vector (resp. convex) bornology on M, which is the coarsest vector (resp. convex) bornology on M such that the canonical embedding J M X : M → ( X , G ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2239.tif"/> is locally bounded. This vector (resp. convex) bornology on M is called the relative vector bornology induced by G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2240.tif"/> , and denoted by G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2241.tif"/> _M. The pair (M, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2242.tif"/> _M) is called a bornological subspace of (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2243.tif"/> ), and members in G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2244.tif"/> _M are said to be relatively bounded in M. It is clear that if G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2245.tif"/> is separated then so is G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2246.tif"/> _M.

13.8 Definition.

Let (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2247.tif"/> ) be a BVS (resp. CBVS), let M be a vector subspace of X and Q_M : X → X/M the quotient map. Then the family Q M ( G ) = { Q M ( B ) : B ∈ G } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2248.tif"/>

188is a base (since Q_M is onto) for a vector (resp. convex) bornology on X/M, called the quotient bornology and denoted by G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2249.tif"/> /M. The pair (X/M, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2250.tif"/> /M) is called the bornological quotient space.

Remark

The quotient bornology on X/M is the finest vector (resp. convex) bornology for which Q_M : (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2251.tif"/> ) → X/M is locally bounded.

13.c

Let (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2252.tif"/> ) be BVS and M a vector subspace of X. Then the quotient bornology G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2253.tif"/> /M on X/M is separated if and only if M is b–closed (for G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2254.tif"/> ).

13.d Complete convex bornologies:

A disk B in a vector space X is said to be completant (or infracomplete) if the seminormed space (X(B), r_B) is a Banach space. A CBVS (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2255.tif"/> ) is called a complete convex bornological vector space (complete CBVS or CCBVS for short) if G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2256.tif"/> admits a basis consisting of completant disks; in this case, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2257.tif"/> is referred to as a complete convex bornology.

(i) Every complete convex bornology must be separated.

(ii) Every b–closed vector subspace M of a CCBVS (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2258.tif"/> ) must be complete w.r.t the relative bornology; and every vector subspace N of a separated CBVS (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2259.tif"/> ), which is complete for the relative bornology, must be b–closed.

(iii) The bornological quotient space of a complete CBVS by a b–closed vector subspace must be a complete CBVS.

(iv) The bornological product space of a family of complete CBVS is a complete CBVS.