Weak topology

In mathematics, weak topology is an alternative term for certain initial topologies on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most used for the initial topology of a topological vector space with respect to its continuous dual; the remainder of this article will deal with this case, one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed if they are closed with respect to the weak topology. Functions are sometimes called weakly continuous if they are continuous with respect to the weak topology. Let K be a topological field, namely a field with a topology such that addition and division are continuous. In most applications K will be either the field of complex numbers or the field of real numbers with the familiar topologies. Let X be a topological vector space over K. Namely, X is a K vector space equipped with a topology so that vector addition and scalar multiplication are continuous.

We may define a different topology on X using the continuous dual space X*. The topological dual space consists of all linear functions from X into the base field K that are continuous with respect to the given topology; the weak topology on X is the initial topology with respect to the family X*. In other words, it is the coarsest topology on X such that each element of X* remains a continuous function. In order to distinguish the weak topology from the original topology on X, the original topology is called the strong topology. A subbase for the weak topology is the collection of sets of the form φ−1 where φ ∈ X* and U is an open subset of the base field K. In other words, a subset of X is open in the weak topology if and only if it can be written as a union of sets, each of, an intersection of finitely many sets of the form φ−1. More if F is a subset of the algebraic dual space the initial topology of X with respect to F, denoted by σ, is the weak topology with respect to F. If one takes F to be the whole continuous dual space of X the weak topology with respect to F coincides with the weak topology defined above.

If the field K has an absolute value | ⋅ | the weak topology σ is induced by the family of seminorms, ‖ x ‖ f = def | f | for all f∈F and x∈X. In particular, weak topologies are locally convex. From this point of view, the weak topology is the coarsest polar topology. If F is a vector space of linear functionals on X which separates points of X the continuous dual of X with respect to the topology σ is equal to F; the weak topology is characterized by the following condition: a net in X converges in the weak topology to the element x of X if and only if φ converges to φ in R or C for all φ in X*. In particular, if xn is a sequence in X xn converges weakly to x if φ → φ as n → ∞ for all φ ∈ X ∗. In this case, it is customary to write x. If X is equipped with the weak topology addition and scalar multiplication remain continuous operations, X is a locally convex topological vector space. If X is a normed space the dual space X* is itself a normed vector space by using the norm ǁφǁ = supǁxǁ≤1|φ|.

This norm gives rise to a topology, called the strong topology, on X*. This is the topology of uniform convergence; the uniform and strong topologies are different for other spaces of linear maps. A space X can be embedded into its double dual X** by x ↦ T x where T x = ϕ, thus T: X → X** is an injective linear mapping, though not surjective. The weak-* topology on X* is the weak topology induced by the image of T: T ⊂ X**. In other words, it is the coarsest topology such that the maps Tx, defined by Tx = φ from X* to the base field R or C remain continuous. A net ϕ λ in X* is convergent to ϕ in the weak-* topology if it converges pointwise: ϕ λ → ϕ for all x in X. In particular, a sequence of ϕ n ∈ X* converges to ϕ provided that ϕ n

Avraham Qanaï

Avraham Ben-Raḥamiël Qanaï is the leader of the Congregation Oraḥ Ṣaddiqim in Albany, New York, a proponent of Karaite Judaism in the United States. He is considered a Ḥakham in the Karaite Jewish community. Qanaï was, from 2005 until 2010, branch secretary of the James Connolly Upstate New York Regional GMB, Industrial Workers of the World of Albany, New York, his father was named Raḥamiël Ben-Yosef Bölekçan. He is a co-author of the book An Introduction to Karaite Judaism: History, Theology and Custom, which discusses historical and modern Karaite Judaism; as well as editor of numerous Karaite texts published by al-Qirqisani Center for the Promotion of Karaite Studies, translator of numerous Karaite texts, including the Karaite Haggadah, © 2000 & 2003, The Abbreviated Shabbat Prayer-book According to the Custom of the Karaite Jews, research papers, including A Hebrew Poem by Hakham Shabetai Ben Mordekhai Tiro and Crimean Karaite Jewish Identity in the 20th Century (published in Eastern European Karaites in the Last Generations, ed. D. Y Shapira and Daniel J. Lasker ISBN 978-965-235-156-2.

Karaite Jewish Congregation Orah Saddiqim al-Qirqisani Center Publications


Calycophyllum is a genus of flowering plants in the family Rubiaceae. It was described by Augustin Pyramus de Candolle in 1830; the genus is found from Central America, South America and the West Indies. Calycophyllum candidissimum DC. common names: Lemonwood, Digame Lancewood - Mexico, Central America, Trinidad, Colombia Calycophyllum intonsum Steyerm. - Venezuela, Brazil Calycophyllum megistocaulum C. M. Taylor - Bolivia, Ecuador, Perú, Brazil Calycophyllum merumense Steyerm. - Guyana Calycophyllum multiflorum Griseb. - Bolivia, Argentina, Paraguay Calycophyllum obovatum Ducke - Guyana, Colombia, Brazil Calycophyllum papillosum J. H. Kirkbr. - Brazil Calycophyllum spectabile Steyerm. - Guyana Calycophyllum spruceanum Hook.f. Ex K. Schum. - Bolivia, Ecuador, Perú, Brazil Calycophyllum tefense J. H. Kirkbr. - Brazil Calycophyllum venezuelense Steyerm. - Venezuela, Guyana Media related to Calycophyllum at Wikimedia Commons Data related to Calycophyllum at Wikispecies Calycophyllum in the World Checklist of Rubiaceae