Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.

Then X is compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ).

# Every infinite subset of X has a complete accumulation point.

A metric space X is complete if and only if every decreasing sequence of non-empty closed subsets of X, with diameters tending to 0, has a non-empty intersection: if F < sub > n </ sub > is closed and non-empty, for every n, and diam ( F < sub > n </ sub >) → 0, then there is a point x ∈ X common to all sets F < sub > n </ sub >.

If X is a set and M is a complete metric space, then the set B ( X, M ) of all bounded functions ƒ from X to M is a complete metric space.

If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.

The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention.

Inevitably, the surviving evidence is not complete enough to determine whether one should interpret, with older scholars, that he wisely curtailed the activities of the Roman Empire to a careful minimum, or perhaps that he was uninterested in events away from Rome and Italy and his inaction contributed to the pressing troubles that faced not only Marcus Aurelius but also the emperors of the third century.

The most complete existing works dealing with the mythical origins of the constellations are by the Hellenistic writer termed pseudo-Eratosthenes and an early Roman writer styled pseudo-Hyginus.

The early Roman calendar, which had no way of designating the days of the winter months other than to lump them together as " winter ", is an example of an incomplete calendar, while the Gregorian calendar is an example of a complete calendar.

Only a storm, which separated the combatants, saved the Roman forces from complete annihilation.

Suffering a complete defeat at the Battle of Chersonesus, the Eastern Roman Emperor Theodosius II gave in to Hun demands and the Peace of Anatolius was signed in autumn 443.

Justinian achieved lasting fame through his judicial reforms, particularly through the complete revision of all Roman law, something that had not previously been attempted.

The Beneventan rite appears to have been less complete, less systematic, and more liturgically flexible than the Roman rite.

A half-hearted attempt was made to complete the temple during Augustus ' reign as the first Roman emperor, but it was not until the accession of Hadrian in the 2nd century AD that the project was finally completed around 638 years after it had begun.

Underneath letters by St. Jerome and Gennadius was the almost complete text of the Institutes of Gaius, probably the first student's textbook on Roman law.

This type of language is used, according to E. Segal, for “ the forceful inversion, the reduction of the master to an abject position of supplication … the master-as-suppliant is thus an extremely important feature of the Plautine comic finale .” The imperative mood is therefore used in the complete role-reversal of the normal relationship between slave and master, and “ those who enjoy authority and respect in the ordinary Roman world are unseated, ridiculed, while the lowliest members of society mount to their pedestals … the humble are in face exalted ”.

Aside from the complete rebuilding of the ancient church of Santa Maria in Trastevere, which boldly features Ionic capitals from former colonnades in the Baths of Caracalla and other richly detailed spolia from Roman monuments, the remaining years of this Pope's life were almost as barren of permanent political results as the first had been.

A dictator would have complete authority over civil and military matters within the Roman imperium, and was not legally responsible for his actions as a dictator and therefore was unquestionable.

By the beginning of the 1st Century BC, the Rex Sacrorum was all but forgotten and the pontifex maximus given almost complete religious authority over the Roman religion.

A complete Roman legion with auxiliaries was annihilated.

In 1512 he became prochancellor at the university and from that time until his death he was in complete control of the destinies of Ingolstadt, on which he impressed the character of ultracatholicism, which made it a bulwark of Roman Catholicism in Germany at that time.

A History of Belgium from the Roman Invasion to the Present Day ( 1921 ) 357 pages ; complete text online

Turning to the Roman aristocracy, in particular the Frangipani family, he gave them the fortress of the Circus Maximus on 31 January 1145, allowing them complete control of the southern portion of the Palatine Hill.

His 37-year rule, the longest of a mature Swedish king so far ( subsequently passed by Gustav V and current Carl XVI Gustav ) saw a complete break with not only the Danish supremacy of the Union but also the Roman Catholic Church, whose assets were nationalised, with the Lutheran Church of Sweden established under his personal control.

The agreement provided for the complete dismemberment of Venice's territory in Italy and for its partition among the signatories: Holy Roman Emperor Maximilian I of the Habsburg, was to receive Padua in addition to Verona and other territories.

A complete tile kiln was found in Park Street some six miles ( 10 km ) from Verulamium in the 1970s, and there is a Roman mausoleum near Rothamsted Park five miles ( 8 km ) away.

No complete early Roman tragedy survives, though it was highly regarded in its day ; historians know of three other early tragic playwrights — Quintus Ennius, Marcus Pacuvius and Lucius Accius.

The Thirty-Nine Articles were not intended as a complete statement of the Christian faith, but of the position of the Church of England in relation to the Roman Catholic Church and dissident Protestants.

* The Roman general Publius Cornelius Scipio secures Gades, thus making Roman control of Spain complete.

Matsuo puzzled and grew anxious over the complete passiveness, concluding that he was the butt of a devilish joke.

But a few days after Fred's return he began hemorrhaging and that was the beginning of early and complete disintegration.

He was a learned and brilliant man, one of the best jurists in Europe and with flashes of penetrating insight, and yet in his dealings with other people, particularly when he tried to be ingratiating, he was capable of an abysmal stupidity that can have come only from a complete incomprehension of human nature and human motives.

They, however much they were in disagreement with the late Victorians over the method by which Britain was Germanized, agreed with them that the end result was the complete extinction of the previous Celtic population and civilization.

A recent editorial discussing a labor-management agreement reached between the Southern Pacific Co. and the Order of Railroad Telegraphers has been criticized on the grounds that it was not based on complete information.

The prize was an old-fashioned, woven cloth hammock, complete with cross-top pillow, fringed side pieces, and hooks for hanging.

Despite several years of front-page stories, the average citizen was unable to get a complete picture of McCarthy until he saw on the television screen what the reporters had been seeing all along but had no effective way of communicating.

The work program, as it was originally proposed, was to take five years to complete.

Another recent achievement was the successful development of a method for the complete combustion in a bomb calorimeter of a metal in fluorine when the product is relatively non-volatile.

Cuban S.S.R.: Whatever may have been the setbacks resulting from the unsuccessful attempt of the Cuban rebels to establish a beachhead on the Castro-held mainland last week, there was at least one positive benefit, and that was the clear-cut revelation to the whole world of the complete conversion of Cuba into a Russian-dominated military base.

This was later known to be the result of concentrating the minute amount of complete antibody found in these sera ; ;

when the insoluble fraction was suspended in a volume of saline equal to that of the original serum sample, no complete antibody activity could be detected.

It was nevertheless almost incredible that four years after Yalta there should be a complete split over Germany, with hot heads on both sides planning to use the Germans against their former allies, and with Nazi-minded Germans expecting to recover their power by fighting on one side or the other.

For the use of students and future restorers, a full, day-by-day record was kept of all three undertakings, complete technical reports on what we found and what we did.

If we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.

Milman Parry rigorously defended the observation that the extant Homeric poems are largely formulaic, and was led to postulate that they could be shown entirely formulaic if the complete corpus of Greek epic survived ; ;

Some time later the missing part of the relic was found and the complete inscription, together with other new evidence, fully corroborated the ancient priest's information.

As a natural outgrowth of this approach it was often suggested that the doctor should complete the preparation for painless intercourse by dilating the vagina.

So filled was Mel Chandler with the spirit of Garryowen that after Korea was over, he took on the job of writing the complete history of the regiment.