Hahn–Banach theorem

In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, it shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, has numerous uses in convex geometry; the theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C of continuous functions on an interval was proved earlier by Eduard Helly, a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz; the most general formulation of the theorem needs some preparation. Given a real vector space V, a function f: V → R is called sublinear if Positive homogeneity: f = γ f for all γ ∈ R+, x ∈ V, Subadditivity: f ≤ f + f for all x, y ∈ V.

Every seminorm on V is sublinear. Other sublinear functions can be useful as well Minkowski functionals of convex sets. Hahn–Banach theorem. If p: V → R is a sublinear function, φ: U → R is a linear functional on a linear subspace U ⊆ V, dominated by p on U, i.e. φ ≤ p ∀ x ∈ U there exists a linear extension ψ: V → R of φ to the whole space V, i.e. there exists a linear functional ψ such that ψ = φ ∀ x ∈ U, ψ ≤ p ∀ x ∈ V. Hahn–Banach theorem. Set K = R or C and let V be a K-vector space with a seminorm p: V → R. If φ: U → K is a K-linear functional on a K-linear subspace U of V, dominated by p on U in absolute value, | φ | ≤ p ∀ x ∈ U there exists a linear extension ψ: V → K of φ to the whole space V, i.e. there exists a K-linear functional ψ such that ψ = φ ∀ x ∈ U, | ψ | ≤ p ∀ x ∈ V. In the complex case of the alternate version, the C-linearity assumptions demand, in addition to the assumptions for the real case, that for every vector x ∈ U, we have ix ∈ U and φ = iφ; the extension ψ is in general not uniquely specified by φ and the proof gives no explicit method as to how to find ψ.

The usual proof for the case of an infinite dimensional space V uses Zorn's lemma or, the axiom of choice. It is now known that the ultrafilter lemma, weaker than the axiom of choice, is strong enough, it is possible to relax the subadditivity condition on p, requiring only that: p ≤ | a | p + | b | p, x, y ∈ V, | a | + | b | ≤ 1. It is further possible to relax the positive homogeneity and the subadditivity conditions on p, requiring only that p is convex; this reveals the intimate connection between the Hahn -- convexity. The Mizar project has formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file; the theorem has several important consequences, some of which are sometimes called "Hahn–Banach theorem": If V is a normed vector space with linear subspace U and if φ: U → K is continuous and linear there exists an extension ψ: V → K of φ, continuous and linear and which has the same operator norm as φ. In other words, in the category of normed vector spaces, the space K is an injective object.

If V is a normed vector space with linear subspace U and if z is an element of V not in the closure of U there exists a continuous linear map ψ: V → K with ψ = 0 for all x in U, ψ = 1, ||ψ|| = dist−1. In particular, if V is a normed vector space and z is an element of V there exists a continuous linear map ψ: V → K such that ψ = ||z|| and ||ψ|| ≤ 1; this implies that the natural injection J from a normed space V into its double dual V′′ is isometric. Hahn–Banach separa

Tülay Tuğcu

Tülay Tuğcu is a retired Turkish judge. She was the President of the Constitutional Court of Chief Justice of Turkey, she retired on June 12, 2007. Tuğcu attended TED College for primary and high school. In 1961, she enrolled in Ankara University Faculty of Law and graduated in 1965. After working as a lawyer for 4 years, she passed the exams required to start working at the Council of State as assistant to Council of State. In 1974, she graduated from Institute of "Public Administration of Turkey and the Middle East" in Ankara. In 1982, she was appointed to the senior judicial ost of Investigation at First Department of the Council of State, where she served until 1992. In 1992, Tülay Tuğcu was elected member of Turkish Council of State and started serving at the Sixth Department. After 3 years, she was transferred to the Tenth Department of Council of State and continued serving there. On December 22, 1999, she was appointed as a member of the Constitutional Court by President Ahmet Necdet Sezer among three candidates determined by the General Assembly of State Council.

Thereafter, she was elected President of the Court of Jurisdictional Disputes on January 6, 2004 and as Chief of the Supreme Court on July 25, 2005 consequently. Tuğcu holds two theses in "Extradition of Criminals" and "The Use of Approval Rights of Administration by High Officials" and a translation in "Productivity", she has two children. In one of her messages to the public on Supreme Court’s official website, she said: "... if we want to name the age we are living in, the best can be the "age of communication". In order to be able to adjust to this age, people’s rights to access the information, the sharing of the accessible information and making it common must be provided. Though it has not been arranged in our constitution, the right to access the information is a "sine qua non" of basic rights and freedom. Without doubt, the internet websites of the public institutions that are equipped with the latest and satisfactory information play a crucial role in managing it." Nur Batur's interview with Tuğcu Official Page of Turkish Constitutional Court

Kartini Beach

Kartini Beach is a tourist beach in Bulu, Central Java. This beach is located 2.5 km west of the hall of the Office of the Regent of Jepara. Regions with an area of 3.5 ha of land is a strategic area, because as the sea transportation routes to the National marine park attractions karimunjawa and Panjang Island. In addition Kartini Beach, can not escape from a traditional event called the "th"; this event is the Jepara community cultural events that took place during one day on the 8th of Shawwal or the week after the Eid al-Fitr. Kartini Beach is called "the Baths" located on the west coast Kartini, because it was used as a trusted public baths to cure rheumatic diseases, itching