...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
where the last limit is in the weak topology for probability measures on $\partial \Omega$ (cf. also [[Weak convergence
5 KB (743 words) - 17:01, 1 July 2020
with the usual product topology) is topologically ergodic; equivalently: whenever for every choice of four
5 KB (696 words) - 08:01, 6 June 2020
There is a natural [[Topological structure (topology)|topological structure (topology)]] on $ \Delta ^ {+} $,
namely, the topology of weak convergence (cf. also [[Weak topology|Weak topology]]), where $ F _ {n} \rightarrow F $
12 KB (1,757 words) - 08:07, 6 June 2020
...="top">[a1]</TD> <TD valign="top"> M.J. Greenberg, J.R. Harper, "Algebraic topology, a first course" , Benjamin/Cummings (1981) {{MR|643101}} {{ZBL|0498.55001}
...="top">[a3]</TD> <TD valign="top"> M.J. Greenberg, J.R. Harper, "Algebraic topology, a first course" , Benjamin/Cummings (1981) {{MR|643101}} {{ZBL|0498.55001}
11 KB (1,584 words) - 11:51, 8 April 2023
corresponding topology coincides with the strong topology. The unit
ball $\{f:\|f\|\le 1\}$, considered in the weak topology, is compact.
5 KB (903 words) - 21:31, 3 January 2021
...d fields" C.M. DeWitt (ed.) B.S. DeWitt (ed.) , ''Relativity, groups and topology'' , Gordon & Breach (1964)</TD></TR><TR><TD valign="top">[a4]</TD> <TD
14 KB (2,063 words) - 02:45, 18 July 2022
...manifolds: An example in Algebra Situs" , ''Proc. Conf. in Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, Ma
2 KB (263 words) - 19:52, 5 March 2018
...The class of continuous closed mappings plays an important role in general topology and its applications. Continuous closed compact mappings are called perfect
...><TR><TD valign="top">[3]</TD> <TD valign="top"> R. Engelking, "General topology" , PWN (1977) (Translated from Polish)</TD></TR></table>
4 KB (635 words) - 17:44, 4 June 2020
If the norm-topology is replaced by the weak topology, then one obtains the so-called weak almost-periodic functions: A function
...unctions coincide with the functions that are uniformly continuous in this topology (for the numbers $ \{ \lambda _ {k} \} $
13 KB (1,809 words) - 19:41, 5 June 2020
where $\epsilon(th)/ t \to 0$ as $t\to 0$ in the topology of $Y$ (see also [[Gâteaux variation|Gâteaux variation]]). If the mapping
2 KB (302 words) - 14:28, 15 April 2023
in the weak$^\star$ topology (see [[Convergence of measures]]).
...e measures $\mu_{x,r}$ as in \ref{e:rescaled} converge in the weak$^\star$ topology to the measure $\mu$ of \ref{e:app_tangent} as $r\downarrow 0$.
12 KB (1,962 words) - 17:00, 13 June 2020
...for the above two conditions — (C) for closure finiteness and (W) for weak topology.
d) the topology of $ X $
7 KB (1,077 words) - 06:29, 30 May 2020
...each of which has a certain structure (defined by algebraic operations, a topology, or by an order relation). The general definition of an operator coincides
...nes a continuous mapping from $X$ into $Y$ where $X$ and $Y$ have the weak topology. Compact operators are often called completely-continuous operators. Someti
14 KB (2,265 words) - 17:06, 24 January 2020
the convergence being understood in the topology of $ Y $.
2 KB (339 words) - 14:29, 15 April 2023
...lambda ^ { p } ( M ^ { 1 } ( G ) )$ with respect to the ultraweak operator topology on $\mathcal{L} ( L _ { \text{C} } ^ { p } ( G ) )$.
...G ) )$ is homeomorphic to $P M _ { p } ( G )$ with the ultraweak operator topology on $\mathcal{L} ( L _ { \text{C} } ^ { p } ( G ) )$. As a consequence, for
11 KB (1,698 words) - 07:42, 27 January 2024
...$H ( G )$ denote the space of analytic functions on $G$ equipped with the topology of [[Uniform convergence|uniform convergence]] on compacta, let $\mathcal{B
5 KB (706 words) - 17:03, 1 July 2020
...s applications in general topology (cf. also [[Combinatorial analysis]]; [[Topology, general]]). The existence of Aronszajn trees is a theorem of [[ZFC]] [[set
...ed sets" K. Kunen (ed.) J.E. Vaughan (ed.) , ''Handbook of Set Theoretic Topology'' , North-Holland (1984)</TD></TR>
5 KB (783 words) - 16:49, 14 January 2021
The limit, in the topology of $ Y $,
3 KB (391 words) - 14:29, 15 April 2023
A topology $ \tau $
with an indefinite metric one cannot specify more than one Fréchet topology subordinate to $ G $,
12 KB (1,807 words) - 08:22, 6 June 2020
...erator ergodic theorem — and its generalizations), in the uniform operator topology (uniform ergodic theorems, see {{Cite|HP}}, {{Cite|DS}}, {{Cite|N}}), while
5 KB (685 words) - 16:32, 6 January 2024
endowed with the weak topology, is compact and is called the spectrum of the algebra $ A $,
3 KB (461 words) - 16:43, 4 June 2020
...e also uses ordered vector spaces on which there is also defined a certain topology compatible with the order. The simplest and most important example of such
...gy. For normed spaces, normality of the cone in the weak and in the strong topology are equivalent.
43 KB (5,981 words) - 17:13, 2 May 2020
...r short, a Parovichenko space). This theorem had wide applications both in topology as well as in the theory of Boolean algebras. The method of proof of this r
...4]</TD> <TD valign="top"> K. Kunen, "Weak P-points in $\mathbf{N}$" , ''Topology, Colloq. Math. Soc. János Bolyai'' , '''23''' (1980) pp. 741–749</TD><
11 KB (1,671 words) - 21:29, 19 November 2017
...uous in the weak (or strong, which is the same in the given case) operator topology. By Stone's theorem, $ U ( t) = e ^ {iAt} $,
3 KB (415 words) - 15:28, 28 February 2022
...a topology (cf. [[Topological structure (topology)|Topological structure (topology)]]) that is compatible with the vector space structure, that is, the follow
with a topology $ \tau $
41 KB (6,085 words) - 08:26, 6 June 2020
...same element of $\mathfrak{A}$. If $\mathfrak{A}$ is endowed with the weak topology (see ), then b) means that the mapping $\R\to\mathfrak{A}$ that takes $t$ t
3 KB (480 words) - 02:51, 15 February 2024
...\overline{G_S G_k} = G_{\mathbf{A}}$? (Here the bar denotes closure in the topology of $G_{\mathbf{A}}$.) If $S = \infty$ ($\infty$ is the set of all Archimede
11 KB (1,671 words) - 18:19, 24 May 2019
...[a8]</TD> <TD valign="top"> D. Sullivan, "Infinitesimal calculations in topology" ''Publ. Math. IHES'' , '''47''' (1977) pp. 269–331</TD></TR></table>
31 KB (4,636 words) - 12:07, 15 December 2019
is a continuous mapping in the topology induced on $ D ^ + $
is valid only in the sense of the weak- topology $ \sigma ( X ^ \prime , X ) $.
34 KB (5,024 words) - 09:12, 21 January 2024
...weak convergence, and in the semi-group of positive-definite functions the topology of uniform convergence on bounded sets.
8 KB (1,162 words) - 19:58, 19 January 2024
...inted out that many theorems are directly related to the geometry and the topology of Banach spaces.
...ar functionals in Banach spaces ultimately evolved to the concept of weak topology. The theory of Banach spaces is a thoroughly studied branch of functional
25 KB (4,253 words) - 23:54, 13 April 2019
converges weakly (see [[#References|[a10]]] and [[Weak topology|Weak topology]]) to $ B _ {P} ( C ) $,
9 KB (1,327 words) - 19:37, 5 June 2020
1) The Lévy metric induces a weak topology in $ {\mathcal F} $(
7 KB (1,007 words) - 04:11, 6 June 2020
...of [[locale]]. The intention was to develop the concept of non-commutative topology introduced by R. Giles and H. Kummer [[#References|[a2]]], while providing
...lign="top"> R. Giles, H. Kummer, "A non-commutative generalization of topology" ''Indiana Univ. Math. J.'' , '''21''' (1971) pp. 91–102</TD></TR>
13 KB (2,065 words) - 18:59, 12 December 2023
...e for the tensor product of $2$-categories, and braid groups" , ''Algebra, Topology, and Category Theory (a collection of papers in honour of Samuel Eilenberg)
18 KB (2,710 words) - 00:41, 15 February 2024
==Topology of $\mathbf{C}$-convex sets.==
...ctions on $E$, endowed with the projective (respectively, inductive) limit topology. An element $\mu$ of the dual space $A ^ { \prime } ( E )$ is called an ana
16 KB (2,533 words) - 09:46, 18 February 2024
...ch case convergence is understood in the sense of the [[Weak topology|weak topology]].
9 KB (1,380 words) - 07:55, 4 March 2022
and the derivative is meant in a topology that is left unspecified. If the mathematical model has been correctly form
in a topology which is the quantum analogue of the probabilistic notion of convergence in
8 KB (1,210 words) - 08:23, 6 June 2020
in the [[Weak topology|weak topology]] $ \sigma ( X , X ^ \prime ) $(
15 KB (2,235 words) - 08:13, 6 June 2020
in the [[Weak topology|weak topology]] induced by $ {\mathcal B} $);
11 KB (1,528 words) - 19:35, 5 June 2020
can be approximated in the strong operator topology by linear operators of finite rank with norm not exceeding 1, and with the
4 KB (651 words) - 15:30, 1 July 2020
The [[Zariski topology|Zariski topology]] on the set of prime ideals (cf. [[Prime ideal|Prime ideal]]) $ \mathop
...ak separation properties (that is, there are non-closed points). A similar topology in the non-commutative case can be introduced on the set $ \mathop{\rm S
7 KB (1,160 words) - 08:00, 6 June 2020
...erator topology follows its continuity with respect to the strong operator topology; for a unitary representation one can define the operation of tensor produc
...1 at the unit element of the group, the [[Topology of uniform convergence|topology of uniform convergence]] on compact subsets of $ G $
24 KB (3,516 words) - 08:27, 6 June 2020
...a group of automorphisms that is continuous with respect to the ultraweak topology on $\mathcal{X}$. Since these groups are isometric, in this discussion it i
...G }$ is an isometric representation of $G$ that is continuous in the weak topology, then for each finite regular [[Borel measure|Borel measure]] $\mu$ on $G$
14 KB (2,151 words) - 17:43, 1 July 2020
with the weak topology. For any real $ \lambda $,
This subset is endowed with a Polish (or Suslin) topology such that any $ u \in U $
8 KB (1,190 words) - 08:16, 20 January 2024
be their duals endowed with the strong topology, let $ L ( E, F ) $
in the topology of uniform convergence on bounded sets. Thus, in this topology, the nuclear operator $ A $
24 KB (3,574 words) - 18:23, 21 January 2021
...es are usually distinguished in variational calculus — a strong and a weak topology and, correspondingly, one defines strong and weak extrema. For instance, as
...iational calculus and the qualitative theory of differential equations and topology. The development of functional analysis made a substantial contribution to
28 KB (4,182 words) - 07:56, 16 April 2023
...clopediaofmath.org/legacyimages/g/g043/g043810/g04381016.png" />, with the
topology given by the countable set of norms
...encyclopediaofmath.org/legacyimages/g/g043/g043810/g043810337.png" />. The
topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
74 KB (9,823 words) - 19:33, 9 November 2014
...thcal{L} _ { \mathbf{C} } ^ { 1 } ( G ) )$ coincides with the compact-open topology on $G$ ([[#References|[a3]]]; see also [[#References|[a9]]], [[#References|
7 KB (1,059 words) - 15:30, 1 July 2020
Low-dimensional topology enters bicategory theory from two dual directions. The commutative diagrams
...al group theory" R. Brown (ed.) T.L. Thickstun (ed.) , ''Low dimensional topology'' , ''Lecture Notes London Math. Soc.'' , '''48''' , Cambridge Univ. Press
24 KB (3,338 words) - 17:29, 7 February 2011