Banach space

In mathematics, more in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit, within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Banach spaces grew out of the study of function spaces by Hilbert, Fréchet, Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are Banach spaces. A Banach space is a vector space X over any scalar field K, equipped with a norm ‖ ⋅ ‖ X and, complete with respect to the distance function induced by the norm, to say, for every Cauchy sequence in X, there exists an element x in X such that lim n → ∞ x n = x, or equivalently: lim n → ∞ ‖ x n − x ‖ X = 0.

The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space X is a Banach space if and only if each convergent series in X converges in X, ∑ n = 1 ∞ ‖ v n ‖ X < ∞ implies that ∑ n = 1 ∞ v n converges in X. Completeness of a normed space is preserved if the given norm is replaced by an equivalent one. All norms on a finite-dimensional vector space are equivalent; every finite-dimensional normed space over R or C is a Banach space. If X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B. In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space X to another normed space is continuous if and only if it is bounded on the closed unit ball of X. Thus, the vector space B can be given the operator norm ‖ T ‖ = sup. For Y a Banach space, the space B is a Banach space with respect to this norm. If X is a Banach space, the space B = B forms a unital Banach algebra.

If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T: X → Y such that T and its inverse T −1 are continuous. If one of the two spaces X or Y is complete so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, i.e. ||T|| = ||x|| for every x in X. The Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ; every normed space X can be isometrically embedded in a Banach space. More for every normed space X, there exist a Banach space Y and a mapping T: X → Y such that T is an isometric mapping and T is dense in Y. If Z is another Banach space such that there is an isometric isomorphism from X onto a dense subset of Z Z is isometrically isomorphic to Y; this Banach space Y is the completion of the normed space X. The underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y.

The completion of X is denoted by X ^. The cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ‖ ‖ 1 = ‖ x ‖ + ‖ y ‖, ‖ ‖ ∞ = max and give rise to isomorphic normed spaces. In this sense, the product X × Y is only if the two factors are complete. If M is a closed linear subspace of a normed space X, there is a natural norm on the quotient space X / M, ‖ x + M ‖ = inf m ∈ M ‖ x + m ‖; the quotient X / M is a Banach space when X is complete. The quotient map from X

Richard Norland

Richard Boyce Norland is an American diplomat. He serves as the United States Ambassador to Libya. Ambassador Richard Norland served as the Foreign Policy Advisor to the Chairman of the Joint Chiefs of Staff, General Joseph Dunford. Prior to that he served as U. S. Ambassador to Georgia, Deputy Commandant/International Affairs Advisor at the National War College, U. S. Ambassador to Uzbekistan, Deputy Chief of Mission at the American Embassy in Kabul and Riga, Latvia. From October 2002 through January 2003, Richard Norland served in Mazar-e-Sharif, Afghanistan as a diplomat with the U. S. Army Civil Affairs team promoting political and economic reconstruction. Richard Norland was Director for European Affairs at the National Security Council for two years during the Clinton and Bush administrations, focusing in particular on the Northern Ireland peace process, as well as on the Baltic States, OSCE, a number of key European partners, he served as Political Counselor at the American Embassy in Dublin, Ireland from 1995 through the negotiation of the 1998 Good Friday Agreement.

Richard Norland served from 1988-1990 as Political Officer at the U. S. Embassy in Moscow, USSR during President Gorbachev's tenure and the period of glasnost and perestroika, he was subsequently detailed to the Pentagon's Office of the Secretary of Defense, where he worked on policy issues following the break-up of the Soviet Union. He served in 1993 as the U. S. representative and acting mission head on the CSCE Mission to Georgia, addressing conflicts in South Ossetia and Abkhazia, visited Chechnya in a similar capacity. Earlier in his career, Richard Norland served in the United States' northernmost diplomatic office, 250 miles north of the Arctic Circle, as Chief of the U. S. Information Office in Tromsø, Norway, he served as Senior Arctic Official coordinating the U. S. chairmanship of the Arctic Council. He was a Special Assistant to the Under Secretary for Political Affairs, he served as Norway-Denmark desk officer, as assistant desk officer for South Africa. His first tour was in Bahrain.

On April 2, 2019, President Donald Trump nominated Richard Norland to be the United States Ambassador to Libya. On August 1, 2019, the Senate confirmed his nomination by voice vote, he assumed office on August 8, 2019. The son of an American diplomat, Ambassador Norland was born in Morocco and grew up in Africa and Europe as well as the United States. Prior to joining the Foreign Service in 1980, Ambassador Norland worked as a legislative analyst in the Iowa House of Representatives, he graduated from Georgetown University's School of Foreign Service in 1977. He has master's degrees from the Johns Hopkins University School of Advanced International Studies and the National War College. In addition to English, he speaks Russian and Norwegian, he and his wife, Mary Hartnett, have two children. United States Ambassador to Georgia Georgia–United States relations


Durmenach is a commune in the Haut-Rhin department in Alsace in north-eastern France. Durmenach is a typical little village located in the South of the region Alsace, it is one of the 120 villages. With the Treaty of Westphalia in 1648, Habsburg domination ceased and Durmenach became French; the village was an important Jewish settlement in the 15th century. Most of the houses in the centre were built by Jewish families between the 18th centuries. In 1826, the Jews still lived in 66 different houses. Durmenach still had 650 Jews out of 1,000 inhabitants at that time. In 1846, the Jewish population represented more than 56% of village. On February 29, 1848, the last antisemitic pogrom in France took place and it happened in the village and its surroundings, it is called Juden Rumpel or Judenrumpell. 75 Jewish houses were burned. An odonym commemorates this event. After 1940, most Jews did not return after the Liberation; the Jewish cemetery of Durmenach dates from 1794 and at the time contained a thousand tombs, 300 of which are still visible.

Communes of the Haut-Rhin department INSEE Durmenach - Official Website