Locally convex topological vector space

In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, a topology can be defined in terms of that family. Although in general such spaces are not normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex spaces that are metrizable, they are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1902 PhD thesis Sur quelques points du calcul fonctionnel.

After the notion of a general topological space was defined by Felix Hausdorff in 1914, although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces. In 1935 von Neumann introduced the general definition of a locally convex space. A notable example of a result which had to wait for the development and dissemination of general locally convex spaces to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spaces. Suppose V is a vector space over a subfield of the complex numbers. A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms. A subset C in V is called Convex if for all x, y in C, 0 ≤ t ≤ 1, tx + y is in C. In other words, C contains all line segments between points in C.

Circled if for all x in C, λx is in C if |λ| = 1. If K = R, this means. For K = C, it means for any x in C, C contains the circle through x, centred on the origin, in the one-dimensional complex subspace generated by x. A cone if for all x in C and 0 ≤ λ ≤ 1, λx is in C. Balanced if for all x in C, λx is in C if |λ| ≤ 1. If K = R, this means that if x is in C, C contains the line segment between x and −x. For K = C, it means for any x in C, C contains the disk with x on its boundary, centred on the origin, in the one-dimensional complex subspace generated by x. Equivalently, a balanced set is a circled cone. Absorbent or absorbing if for every x in V, there exists r > 0 such that x is in tC for all t ∈ K satisfying |t| > r. The set C can be scaled out by any "large" value to absorb every point in the space. Convex if it is both balanced and convex. More succinctly, a subset of V is convex if it is closed under linear combinations whose coefficients sum to ≤ 1; such a set is absorbent if it spans all of V.

Definition. A topological vector space is called locally convex if the origin has a local base of convex absorbent sets; because translation is continuous, all translations are homeomorphisms, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector. A seminorm on V is a map p: V → R such that p is positive or positive semidefinite: p ≥ 0. P is positive homogeneous or positive scalable: p = |λ| p for every scalar λ. So, in particular, p = 0. P is subadditive, it satisfies the triangle inequality: p ≤ p + p. If p satisfies positive definiteness, which states that if p = 0 x = 0 p is a norm. While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms, defined below. Definition. A locally convex space is defined to be a vector space V along with a family of seminorms α ∈ A on V. A locally convex space carries a natural topology, called the initial topology induced by the seminorms. By definition, it is the coarsest topology for which all the mappings { p α, y: V → R x ↦ p α y ∈ V, α ∈ A are continuous.

A base of neighborhoods of y for this topology is obtained in the following way: for every finite subset B of A and every ε > 0, let U B, ε =. {\displaystyle U_=\{x\in V:p_<\varepsilon \ \forall \alpha \in B

Lin Yue

Lin Yue. He competed for Team China in Diving at the 2008 Summer Olympics in Beijing and won gold with Huo Liang in the Men's synchronized 10 metre platform, he won gold in the same event at the 2016 Summer Olympics in Rio with Chen Aisen. Lin Yue started diving when he was little, training in a sporting school in his hometown, with a coach named Li Huansen. In the following five years, he went to Guangdong Provincial Weilun Sporting School, under instructor Cao Ke. In the year 2004, he was selected in the Beijing Diving team, his current coach is Zhong Shaozhen. He claimed the bronze medal in the 2007 World Aquatics Championships, he claimed the gold medal at the 2008 World Cup - 10m platform synchro. He won gold medals in the Men's synchronized 10m platform at the 2008 Summer Olympics in Beijing, the 2009 ROMA FINA World Championships, the 2016 Summer Olympics in Rio, he became the first diver to win a second Olympic gold medal in this event

Synod of Kells

The Synod of Kells took place in 1152, under the presidency of Giovanni Cardinal Paparoni, continued the process begun at the Synod of Ráth Breasail of reforming the Irish church. The sessions were divided between the abbeys of Kells and Mellifont, in times the synod has been called the Synod of Kells-Mellifont and the Synod of Mellifont-Kells, its main effect was to increase the number of archbishops from two to four, to redefine the number and size of dioceses. The Primacy of Ireland was granted to the Archdiocese of Armagh. Máel Máedóc Ua Morgair was made a priest as vicar to Celsus, his first sees were Down and Connor, he was located at Bangor Abbey. On the death of Celsus in 1129, Malachy was nominated as his successor at Armagh, now the prime see in Ireland. An internal church dispute over the succession and proposals for reform obliged him to concede the position to Gelasius. In 1137, lacking papal confirmation of the appointment of Malachy by Rome asked him to secure the archbishop's pallium at the hands of the Pope or his legate.

Malachy reached Rome but the Pope, Innocent II, would only grant the pallia to Malachy at the request of an Irish National Synod. To facilitate this, he made Malachy his papal legate. Malachy returned to Ireland accompanied by a number of Cistercian monks provided by St. Bernard. In 1148 a synod of bishops was assembled at Inispatric. Malachy died on the way at Clairvaux, France, in November. A synod was summoned to Kells in 1152; this synod approved the consecration of four archbishops. Tairrdelbach Ua Conchobair, the High King of Ireland, approved the decrees, the pallia were conferred by the Papal Legate, Giovanni Cardinal Paparoni. Ireland was divided into thirty-six sees, four metropolitan sees: Armagh, Cashel and Dublin. Armagh was granted Primacy; the diocese of Dublin, ruled by the Ostmen, was united with Glendalough. Gregory, the incumbent bishop, accepted Ostman separatism came to an end; the diocesan system was further reorganised, with the number of metropolitan provinces being increased from two to four, by raising the dioceses of Dublin and Tuam to archdioceses.

The four provinces of Armagh, Cashel and Tuam corresponded to the contemporary boundaries of the provinces of Ulster, Munster and Connacht respectively. In most cases the dioceses corresponded with the territories controlled by the Irish clans, the clan chiefs liked to appoint family members as bishops and church officials; the diocesan structure established by the synod survived until the sixteenth century, still forms the basis of the territorial structure of both the Catholic Church and the Church of Ireland, with many of the sees now merged. Ardagh: reduced in size by creation of Diocese of Kells Armagh Clonard: confirmed as see for East Meath Connor Dar-Luis: status of area uncertain Down Duleek Kells: established as see for the Kingdom of Breifne. Absorbed by Diocese of Meath in 1211 Louth: see moved from Clogher and area extended at the expense of Armagh. See returned to Clogher by 1192 Maghera: see transferred to Derry in 1254 Raphoe: created in the late 12th century subsequent to the synod Ardfert: lost territory to Scattery Island Cashel Cloyne: formed from part of Cork Cork: lost territory to Cloyne and Ross Emly Kilfenora: formed from part of Diocese of Killaloe.

Incorporated into Limerick by end of 12th century Waterford: lost territory to create Lismore Dublin Ferns Glendalough: united to Dublin in 1216 Kildare Kilkenny Leighlin Achonry Clonfert Killala Kilmacduagh Mayo: merged with Tuam 1209 Roscommon moved to Elphin 1156 Tuam Annaghdown was created circa 1179 Synod of Ráth Breasail Synod of Cashel Peter Galloway, The Cathedrals of Ireland, Belfast 1992 Healy, John. "Synod of Kells". The ancient Irish church. London: Religious Tract Society. Pp. 176–180. Geoffrey Keating. Foras Feasa Book I–II Geoffrey Keating. Http:// The History of Ireland The Dioceses of Ireland, Territorial History