...us at this point, then the subdifferential is non-empty and compact in the topology $ \sigma ( Y, X) $.
that is compact in the topology $ \sigma ( Y, X) $
3 KB (480 words) - 14:55, 7 June 2020
$#C+1 = 49 : ~/encyclopedia/old_files/data/S090/S.0900610 Strong topology
The [[Weak topology|weak topology]] on $ L $
3 KB (419 words) - 08:24, 6 June 2020
A concept in measure theory, determined by a certain topology in a space of measures that are defined on a certain [[Algebra of sets|σ-a
===The norm or [[strong topology]]===
9 KB (1,371 words) - 08:33, 16 August 2013
...e often used: the strong topology determined by this norm and the weak-$*$-topology.
...uses the term dual space. The weak-$*$-topology on $E^{*}$ is the weakest topology on $E^{*}$ for which all the evaluation mappings $f\mapsto f(x)$, $f\in E^{
1 KB (186 words) - 18:01, 30 November 2012
...space]] which is boundedly compact (in itself) in the weak (resp. strong) topology is [[Reflexive space|reflexive]] (resp. a [[Montel space]]). A normed space
1 KB (200 words) - 17:07, 17 October 2014
$#C+1 = 64 : ~/encyclopedia/old_files/data/O068/O.0608410 Operator topology
A topology on the space $ L( E, F ) $
5 KB (737 words) - 08:04, 6 June 2020
that are continuous in the weak topology of $ U $;
cf. [[Weak topology|Weak topology]]); and let the set $ X = \{ {x } : {f _ {i} ( x) \geq 0, i = 1 \dots m
6 KB (879 words) - 19:35, 12 January 2024
...mplete Mackey space and the space dual to $E$ equipped with the strong $E$-topology is semi-reflexive, then $E$ is reflexive.
3 KB (444 words) - 19:36, 11 October 2023
The [[locally convex topology]] on a vector space $X$ generated by the family of [[semi-norm]]s $p(x)=|f(
...y as introduced above is often denoted by $\sigma(X,F)$. It is a Hausdorff topology if and only if $F$ is a [[total set]], that is, separates the points of $X$
1 KB (150 words) - 22:17, 10 December 2016
...a complete set in the weak topology will be a complete set in the initial topology also.
1 KB (254 words) - 09:44, 5 August 2014
...$X$ that is homeomorphic to a subset of a [[Banach space]] with the [[weak topology]] [[#References|[a3]]].
...a subset of $c_0(I)$ in the weak topology (or, equivalently, the pointwise topology), for some set $I$; and iii) $X$ has a family $\mathcal{B} = \cup_{n<\omega
2 KB (383 words) - 17:56, 31 December 2017
$#C+1 = 38 : ~/encyclopedia/old_files/data/V096/V.0906000 Vague topology
...sets|Borel field of sets]]), generated by the (set of open subsets of the) topology of $ X $.
3 KB (507 words) - 19:20, 1 January 2021
...s is equivalent to the space being Lindelöf. If a Banach space in the weak topology is topologically generated by some compactum lying within it, then it is pa
...="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)</TD></
9 KB (1,401 words) - 14:04, 30 July 2014
formed by the continuous functions and endowed with the following Hausdorff topology: a fundamental system of neighbourhoods of zero in $ X $
to be a (topologically-) free set in the weak topology $ \sigma ( X, X ^ {*} ) $
3 KB (399 words) - 19:40, 5 June 2020
...mages/t/t093/t093470/t0934705.png" /> is considered with the weak operator topology. The concept of a totally-irreducible set, initially introduced for Banach
8 KB (1,059 words) - 17:18, 7 February 2011
...] is bounded in the [[strong topology]] and relatively compact in the weak topology.
...ment that the dual of a barrelled space is quasi-complete for any $\sigma$-topology. (For the last notion see [[Topological vector space]]; [[Space of mappings
3 KB (507 words) - 13:44, 17 March 2023
...rongly exposed points, [[#References|[a2]]] (cf. also [[Weak topology|Weak topology]]). R.E. Huff and P.D. Morris [[#References|[a7]]] showed that this propert
...="top"> M.J. Fabian, "Gâteaux differentiability of convex functions and topology-weak Asplund spaces" , Wiley (1997)</td></tr></table>
6 KB (893 words) - 16:59, 1 July 2020
In [[metric space]]s and [[Banach space]]s with the [[weak topology]] the notions of compactness, sequential compactness and countable compactn
...="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)</TD></
1 KB (207 words) - 20:42, 2 November 2014
...p"> K. Kunen, "Weak $P$-points in $\mathbf{N}^*$" Á. Császár (ed.) , ''Topology (Proc. Fourth Colloq., Budapest, 1978)'' , '''II''' , North-Holland (1980)
3 KB (409 words) - 20:01, 21 November 2017
$#C+1 = 27 : ~/encyclopedia/old_files/data/S110/S.1100200 Skorokhod topology
A [[Topological structure (topology)|topological structure (topology)]] on the space $ D [ 0,1 ] $
4 KB (518 words) - 08:14, 6 June 2020
...space-time) is unstable with respect to a (still to be specified) natural topology of space-times [[#References|[a9]]], [[#References|[a10]]]. Here, "future i
5 KB (717 words) - 15:40, 19 February 2021
...TR><TD valign="top">[4]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)</TD></TR></table>
2 KB (272 words) - 13:05, 24 December 2020
...ology compatible with the ordinary topology on all its simplices; the weak topology may serve as an example.
In topology and differential geometry, these complexes occur most frequently as realiza
3 KB (419 words) - 17:16, 7 February 2011
topology $ \sigma ( E ^ \prime , E) $.
topology (the Banach–Alaoglu theorem).
6 KB (887 words) - 08:06, 6 June 2020
...ial role, and often a key one, in constructions in all branches of general topology and in many of its applications. In particular, it is of fundamental import
...role — in particular, in functional analysis (Banach spaces with the weak topology, measures on topological spaces), in the general theory of optimal control,
3 KB (412 words) - 17:06, 7 May 2016
Path-connected spaces play an important role in homotopic topology. If a space $ X $
...TR><TD valign="top">[1]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)</TD></TR></table>
3 KB (480 words) - 10:17, 19 January 2022
...etermined by the nature of these concepts. The initial concepts of general topology are the concepts of a [[topological space]] and a [[continuous mapping]], i
...ures defined on the set of points of the space, in some way related to its topology. The topological invariance of the homology groups may serve as an example.
10 KB (1,522 words) - 19:35, 25 March 2023
that converges weakly (cf. [[Weak topology|Weak topology]]) to a [[Probability measure|probability measure]] $ P $(
3 KB (375 words) - 08:14, 6 June 2020
...at it is possible to construct an equivalent metric (i.e., giving the same topology) under which the space $\mathcal{D}$ becomes a separable and complete metri
...D}$ is called the Skorokhod space (cf. also [[Skorokhod topology|Skorokhod topology]]). This space is very important in the theory of random processes (cf. als
8 KB (1,131 words) - 14:43, 27 January 2024
...–Borel structure contains all sets of the Borel structure generated by the topology of $ \widehat{A} $; each point of $ \widehat{A} $ is a Borel set in the Mac
...isomorphic, as a Borel structure, to the Borel structure generated by the topology of some complete separable metric space).
3 KB (478 words) - 04:32, 4 December 2016
...g on which mappings $X \rightarrow Y$ are included in $F$ and what natural topology $F$ is endowed with. The choice of $F$ is related to the presence of additi
...ing way. A family $S$ of subsets of $X$ is fixed, and a [[pre-base]] for a topology $\mathfrak{T}$ on $F$ is formed by sets of the form
8 KB (1,267 words) - 05:59, 22 April 2023
...teria of reflexivity of a Banach space $X$ is weak compactness (cf. [[Weak topology]]) of the unit ball of this space.
2 KB (354 words) - 18:18, 26 September 2017
...$\mathbf R _ { d}$ be the real line $\mathbf{R}$ endowed with the discrete topology and suppose the algebra $A ( G )$ consists of the functions in $C ( G )$ wh
...ce|Hausdorff space]] upon which the real line $\mathbf{R}$ (with the usual topology) acts as a locally compact [[Transformation group|transformation group]]. T
7 KB (1,114 words) - 19:36, 23 December 2023
...onvergence to topologies on the space which are stronger than the original topology (for references to these results, see the historical survey of [[#Reference
...een made to replace the [[Weak topology|weak topology]] of $X$ by a weaker topology, $\sigma ( X , Y )$, generated by a subspace $Y$ of the dual space of $X$ w
5 KB (714 words) - 15:30, 1 July 2020
...al signed Radon measures and the corresponding topology is called ''narrow topology'' by some authors. Several other notions of convergence can be introduced o
4 KB (739 words) - 10:38, 23 November 2013
The density topology $\mathcal{T}_d$ on $\mathbb R$ consists of the family of all subsets $E\sub
...tion $\mathcal{A} = C (\mathcal{T}_d)$ shows the importance of the density topology in real analysis, since the class $\mathcal{A}$ is strongly tied to the the
6 KB (865 words) - 22:40, 17 August 2013
...city of their definition, makes them important standard objects in general topology. However, the topological structure of the Tikhonov cubes is far from trivi
2 KB (392 words) - 15:51, 11 August 2014
Define a topology on $ X $
This topology is called the hull-kernel topology on $ X $.
7 KB (1,058 words) - 08:29, 6 June 2020
...$ is dense in $T$. If the algebraic dual $E^*$ of $E$, is given the [[weak topology]] (so that $E^* \simeq \prod_{\alpha \in A} K$,where $K$ is the base field
1 KB (185 words) - 19:44, 27 February 2021
...s functions on the interval $[ 0,1 ]$, with the [[Uniform topology|uniform topology]], the weak convergence
6 KB (857 words) - 21:45, 15 December 2020
...]] on $X$ generated by the family of all superharmonic functions. This new topology was introduced in classical potential theory by Brelot and H. Cartan around
...and $V$ are open in the fine topology, $U$ is a Borel set in the original topology of $X$ and $x \in V \subset U \subset X$.
6 KB (870 words) - 16:57, 1 July 2020
...in the strong operator topology actually converges in the uniform operator topology (see [[#References|[a3]]], [[#References|[a7]]]). In particular, this impli
6 KB (875 words) - 20:01, 27 February 2021
is continuous in the strong topology and sequentially continuous in the weak $ * $
topology of $ \mathfrak X ^ {*} $;
7 KB (1,031 words) - 16:43, 4 June 2020
algebras (weakly Borel, Borel, etc.) that are naturally connected with the topology of $ E $;
endowed with the weak-* topology (so that $ E ^ \prime = V $),
7 KB (1,017 words) - 08:00, 6 June 2020
...the [[Mackey topology]] $\tau(X,X^\alpha)$, or the [[Normal space|normal]] topology is taken. If $X \supset \phi$ and $Y$ are [[BK-space]]s (i.e., Banach [[FK-
3 KB (503 words) - 17:02, 4 October 2017
in the natural weak or uniform topology of the space $ \mathfrak A $
provided with the weak topology, may be considerably strengthened: For any monotone sequence $ f(n) > 0 $
8 KB (1,151 words) - 12:12, 21 March 2022
...respect to the relative $\sigma(F',F)$- (weak) topology on $D(T')$ and the topology on $E'$ of uniform convergence on $\sigma(E,E')$-$K$-convergent sequences.
4 KB (671 words) - 16:06, 20 January 2021
...h space]] possessing a weakly compact subset $K$ (cf. [[Weak topology|Weak topology]]) whose [[linear span]] is dense. These spaces have regularity properties
...ontinuous in the order topology of $[ \omega _ { 0 } , \mu ]$ and the norm topology of $X$, for each $x \in X$. Properties of Banach spaces admitting a project
16 KB (2,514 words) - 19:53, 23 December 2023
|valign="top"|{{Ref|Sp}}||valign="top"| E.H. Spanier, "Algebraic topology", McGraw-Hill (1966) pp. Chapt. 2, §2; Chapt. 7, §2 {{MR|0210112}} {{M
1 KB (187 words) - 22:36, 24 November 2013
...is sufficiently rich and is dense in the set of all distributions with the topology of weak convergence.
2 KB (258 words) - 14:33, 7 October 2014