Civil Rights
Movements, leaders, victories and the continuing fight for equality.
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Movements, leaders, victories and the continuing fight for equality.
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Meet the people whose lives, choices and achievements shaped the journey.
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Wilmington, North Carolina once had a thriving Black middle class and an elected interracial government. In 1898 white supremacists used violence to overthrow that government, kill Black residents and drive many others from the city.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
Open mapping theorem may refer to:
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Open mapping theorem may refer to: Open mapping theorem (functional analysis) (also known as the Banach–Schauder theorem), states that a surjective continuous linear transformation of a Banach space X onto a Banach space Y is an open mapping Open mapping theorem (complex analysis), states that a non-constant holomorphic function on a connected open set in the complex plane is an open mapping Open mapping theorem (topological groups), states that a surjective continuous homomorphism of a locally compact Hausdorff group G onto a locally compact Hausdorff group H is an open mapping if G is σ-compact. Like the open mapping theorem in functional analysis, the proof in the setting of topological groups uses the Baire category theorem.
In complex analysis, the open mapping theorem states that if U {\displaystyle U} is a domain of the complex plane C {\displaystyle \mathbb {C} } and f : U → C {\displaystyle f:U\to \mathbb {C} } is a non-constant holomorphic function, then f {\displaystyle f} is an open map (i.e. it sends open subsets of U {\displaystyle U} to open subsets of C {\displaystyle \mathbb {C} } , and we have invariance of domain.). The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function f ( x ) = x 2 {\displaystyle f(x)=x^{2}} is not an open map, as the image of the open interval ( − 1 , 1 ) {\displaystyle (-1,1)} is the half-open interval [ 0 , 1 ) {\displaystyle [0,1)} . The theorem for example implies that a non-constant holomorphic function cannot map an open disk onto a portion of any line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. A special case is also called the bounded inverse theorem (also called inverse mapping theorem or Banach isomorphism theorem), which states that a bijective bounded linear operator T {\displaystyle T} from one Banach space to another has bounded inverse T − 1 {\displaystyle T^{-1}} .
In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Brown v. Board of Education in 1954.
