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In 1761, Franklin wrote a letter to Mary Stevenson describing his experiments on the relationship between color and heat absorption. He found that darker color clothes got hotter when exposed to sunlight than lighter color clothes, an early demonstration of black body thermal radiation. One experiment he performed consisted of placing square pieces of cloth of various color out in the snow on a sunny day. He waited some time and then measured that the black pieces sank furthest into the snow of all the colors, indicating that they got the hottest and melted the most snow.
According to Michael Faraday, Franklin's experiments on the non-conduction of ice are worth mentioning, although the law of the general effect of liquefaction on electrolytes is not attributed to Franklin. However, as reported in 1836 by Franklin's great-grandson Alexander Dallas Bache of the University of Pennsylvania, the law of the effect of heat on the conduction of bodies otherwise non-conductors, for example, glass, could be attributed to Franklin. Franklin wrote, "... A certain quantity of heat will make some bodies good conductors, that will not otherwise conduct ..." and again, "... And water, though naturally a good conductor, will not conduct well when frozen into ice."
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Oceanography and hydrodynamics.
As deputy postmaster, Franklin became interested in North Atlantic Ocean circulation patterns. While in England in 1768, he heard a complaint from the Colonial Board of Customs. British packet ships carrying mail had taken several weeks longer to reach New York than it took an average merchant ship to reach Newport, Rhode Island. The merchantmen had a longer and more complex voyage because they left from London, while the packets left from Falmouth in Cornwall. Franklin put the question to his cousin Timothy Folger, a Nantucket whaler captain, who told him that merchant ships routinely avoided a strong eastbound mid-ocean current. The mail packet captains sailed dead into it, thus fighting an adverse current of . Franklin worked with Folger and other experienced ship captains, learning enough to chart the current and name it the Gulf Stream, by which it is still known today.
Franklin published his Gulf Stream chart in 1770 in England, where it was ignored. Subsequent versions were printed in France in 1778 and the U.S. in 1786. The British original edition of the chart had been so thoroughly ignored that everyone assumed it was lost forever until Phil Richardson, a Woods Hole oceanographer and Gulf Stream expert, discovered it in the Bibliothèque Nationale in Paris in 1980. This find received front-page coverage in "The New York Times". It took many years for British sea captains to adopt Franklin's advice on navigating the current; once they did, they were able to trim two weeks from their sailing time. In 1853, the oceanographer and cartographer Matthew Fontaine Maury noted that while Franklin charted and codified the Gulf Stream, he did not discover it:
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An aging Franklin accumulated all his oceanographic findings in "Maritime Observations", published by the Philosophical Society's "transactions" in 1786. It contained ideas for sea anchors, catamaran hulls, watertight compartments, shipboard lightning rods and a soup bowl designed to stay stable in stormy weather.
While traveling on a ship, Franklin had observed that the wake of a ship was diminished when the cooks scuttled their greasy water. He studied the effects on a large pond in Clapham Common, London. "I fetched out a cruet of oil and dropt a little of it on the water ... though not more than a teaspoon full, produced an instant calm over a space of several yards square." He later used the trick to "calm the waters" by carrying "a little oil in the hollow joint of [his] cane."
Meteorology.
On October 21, 1743, according to the popular myth, a storm moving from the southwest denied Franklin the opportunity of witnessing a lunar eclipse. He was said to have noted that the prevailing winds were actually from the northeast, contrary to what he had expected. In correspondence with his brother, he learned that the same storm had not reached Boston until after the eclipse, despite the fact that Boston is to the northeast of Philadelphia. He deduced that storms do not always travel in the direction of the prevailing wind, a concept that greatly influenced meteorology. After the Icelandic volcanic eruption of Laki in 1783, and the subsequent harsh European winter of 1784, Franklin made observations on the causal nature of these two seemingly separate events. He wrote about them in a lecture series.
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Population studies.
Franklin had a major influence on the emerging science of demography or population studies. In the 1730s and 1740s, he began taking notes on population growth, finding that the American population had the fastest growth rate on Earth. Emphasizing that population growth depended on food supplies, he emphasized the abundance of food and available farmland in America. He calculated that America's population was doubling every 20 years and would surpass that of England in a century. In 1751, he drafted "Observations concerning the Increase of Mankind, Peopling of Countries, etc." Four years later, it was anonymously printed in Boston and was quickly reproduced in Britain, where it influenced the economist Adam Smith and later the demographer Thomas Malthus, who credited Franklin for discovering a rule of population growth. Franklin's predictions on how British mercantilism was unsustainable alarmed British leaders who did not want to be surpassed by the colonies, so they became more willing to impose restrictions on the colonial economy.
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Kammen (1990) and Drake (2011) say Franklin's "Observations concerning the Increase of Mankind" (1755) stands alongside Ezra Stiles' "Discourse on Christian Union" (1760) as the leading works of 18th-century Anglo-American demography; Drake credits Franklin's "wide readership and prophetic insight." Franklin was also a pioneer in the study of slave demography, as shown in his 1755 essay. In his capacity as a farmer, he wrote at least one critique about the negative consequences of price controls, trade restrictions, and subsidy of the poor. This is succinctly preserved in his letter to the "London Chronicle" published November 29, 1766, titled "On the Price of Corn, and Management of the poor."
Decision-making.
In a 1772 letter to Joseph Priestley, Franklin laid out the earliest known description of the Pro & Con list, a common decision-making technique, now sometimes called a decisional balance sheet:
Views on religion, morality, and slavery.
Like the other advocates of republicanism, Franklin emphasized that the new republic could survive only if the people were virtuous. All his life, he explored the role of civic and personal virtue, as expressed in "Poor Richard's" aphorisms. He felt that organized religion was necessary to keep men good to their fellow men, but rarely attended religious services himself. When he met Voltaire in Paris and asked his fellow member of the Enlightenment vanguard to bless his grandson, Voltaire said in English, "God and Liberty," and added, "this is the only appropriate benediction for the grandson of Monsieur Franklin."
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Franklin's parents were both pious Puritans. The family attended the Old South Church, the most liberal Puritan congregation in Boston, where Benjamin Franklin was baptized in 1706. Franklin's father, a poor chandler, owned a copy of a book, "Bonifacius: Essays to Do Good", by the Puritan preacher and family friend Cotton Mather, which Franklin often cited as a key influence on his life. "If I have been," Franklin wrote to Cotton Mather's son seventy years later, "a useful citizen, the public owes the advantage of it to that book." His first pen name, Silence Dogood, paid homage both to the book and to a widely known sermon by Mather. The book preached the importance of forming voluntary associations to benefit society. Franklin learned about forming do-good associations from Mather, but his organizational skills made him the most influential force in making voluntarism an enduring part of the American ethos.
Franklin formulated a presentation of his beliefs and published it in 1728. He no longer accepted the key Puritan ideas regarding salvation, the divinity of Jesus, or indeed much religious dogma. He classified himself as a deist in his 1771 autobiography, although he still considered himself a Christian. He retained a strong faith in a God as the wellspring of morality and goodness in man, and as a Providential actor in history responsible for American independence.
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At a critical impasse during the Constitutional Convention in June 1787, he attempted to introduce the practice of daily common prayer with these words:
The motion gained almost no support and was never brought to a vote.
Franklin was an enthusiastic admirer of the evangelical minister George Whitefield during the First Great Awakening. He did not himself subscribe to Whitefield's theology, but he admired Whitefield for exhorting people to worship God through good works. He published all of Whitefield's sermons and journals, thereby earning a lot of money and boosting the Great Awakening.
When he stopped attending church, Franklin wrote in his autobiography:
Franklin retained a lifelong commitment to the non-religious Puritan virtues and political values he had grown up with, and through his civic work and publishing, he succeeded in passing these values into the American culture permanently. He had a "passion for virtue." These Puritan values included his devotion to egalitarianism, education, industry, thrift, honesty, temperance, charity and community spirit. Thomas Kidd states, "As an adult, Franklin touted ethical responsibility, industriousness, and benevolence, even as he jettisoned Christian orthodoxy."
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The classical authors read in the Enlightenment period taught an abstract ideal of republican government based on hierarchical social orders of king, aristocracy and commoners. It was widely believed that English liberties relied on their balance of power, but also hierarchal deference to the privileged class. "Puritanism ... and the epidemic evangelism of the mid-eighteenth century, had created challenges to the traditional notions of social stratification" by preaching that the Bible taught all men are equal, that the true value of a man lies in his moral behavior, not his class, and that all men can be saved. Franklin, steeped in Puritanism and an enthusiastic supporter of the evangelical movement, rejected the salvation dogma but embraced the radical notion of egalitarian democracy.
Franklin's commitment to teach these values was itself something he gained from his Puritan upbringing, with its stress on "inculcating virtue and character in themselves and their communities." These Puritan values and the desire to pass them on, were one of his quintessentially American characteristics and helped shape the character of the nation. Max Weber considered Franklin's ethical writings a culmination of the Protestant ethic, which ethic created the social conditions necessary for the birth of capitalism.
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One of his characteristics was his respect, tolerance and promotion of all churches. Referring to his experience in Philadelphia, he wrote in his autobiography, "new Places of worship were continually wanted, and generally erected by voluntary Contribution, my Mite for such purpose, whatever might be the Sect, was never refused." "He helped create a new type of nation that would draw strength from its religious pluralism." The evangelical revivalists who were active mid-century, such as Whitefield, were the greatest advocates of religious freedom, "claiming liberty of conscience to be an 'inalienable right of every rational creature. Whitefield's supporters in Philadelphia, including Franklin, erected "a large, new hall, that ... could provide a pulpit to anyone of any belief." Franklin's rejection of dogma and doctrine and his stress on the God of ethics and morality and civic virtue made him the "prophet of tolerance." He composed "A Parable Against Persecution," an apocryphal 51st chapter of Genesis in which God teaches Abraham the duty of tolerance. While he was living in London in 1774, he was present at the birth of British Unitarianism, attending the inaugural session of the Essex Street Chapel, at which Theophilus Lindsey drew together the first avowedly Unitarian congregation in England; this was somewhat politically risky and pushed religious tolerance to new boundaries, as a denial of the doctrine of the Trinity was illegal until the 1813 Act.
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Although his parents had intended for him a career in the church, Franklin as a young man adopted the Enlightenment religious belief in deism, that God's truths can be found entirely through nature and reason, declaring, "I soon became a thorough Deist." He rejected Christian dogma in a 1725 pamphlet "A Dissertation on Liberty and Necessity, Pleasure and Pain", which he later saw as an embarrassment, while simultaneously asserting that God is "all wise, all good, all powerful." He defended his rejection of religious dogma with these words: "I think opinions should be judged by their influences and effects; and if a man holds none that tend to make him less virtuous or more vicious, it may be concluded that he holds none that are dangerous, which I hope is the case with me." After the disillusioning experience of seeing the decay in his own moral standards, and those of two friends in London whom he had converted to deism, Franklin decided that deism was true but it was not as useful in promoting personal morality as were the controls imposed by organized religion. Ralph Frasca contends that in his later life he can be considered a non-denominational Christian, although he did not believe Christ was divine.
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In a major scholarly study of his religion, Thomas Kidd argues that Franklin believed that true religiosity was a matter of personal morality and civic virtue. Kidd says Franklin maintained his lifelong resistance to orthodox Christianity while arriving finally at a "doctrineless, moralized Christianity." According to David Morgan, Franklin was a proponent of "generic religion." He prayed to "Powerful Goodness" and referred to God as "the infinite." John Adams noted that he was a mirror in which people saw their own religion: "The Catholics thought him almost a Catholic. The Church of England claimed him as one of them. The Presbyterians thought him half a Presbyterian, and the Friends believed him a wet Quaker." Adams himself decided that Franklin best fit among the "Atheists, Deists, and Libertines." Whatever else Franklin was, concludes Morgan, "he was a true champion of generic religion." In a letter to Richard Price, Franklin states that he believes religion should support itself without help from the government, claiming, "When a Religion is good, I conceive that it will support itself; and, when it cannot support itself, and God does not take care to support, so that its Professors are oblig'd to call for the help of the Civil Power, it is a sign, I apprehend, of its being a bad one."
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In 1790, just about a month before he died, Franklin wrote a letter to Ezra Stiles, president of Yale University, who had asked him his views on religion:
On July 4, 1776, Congress appointed a three-member committee composed of Franklin, Jefferson, and Adams to design the Great Seal of the United States. Franklin's proposal (which was not adopted) featured the motto: "Rebellion to Tyrants is Obedience to God" and a scene from the Book of Exodus he took from the frontispiece of the Geneva Bible, with Moses, the Israelites, the pillar of fire, and George III depicted as pharaoh.
The design that was produced was not acted upon by Congress, and the Great Seal's design was not finalized until a third committee was appointed in 1782.
Franklin strongly supported the right to freedom of speech:
Thirteen Virtues.
Franklin sought to cultivate his character by a plan of 13 virtues, which he developed at age 20 (in 1726) and continued to practice in some form for the rest of his life. His autobiography lists his 13 virtues as:
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Franklin did not try to work on them all at once. Instead, he worked on only one each week "leaving all others to their ordinary chance." While he did not adhere completely to the enumerated virtues, and by his own admission he fell short of them many times, he believed the attempt made him a better man, contributing greatly to his success and happiness, which is why in his autobiography, he devoted more pages to this plan than to any other single point and wrote, "I hope, therefore, that some of my descendants may follow the example and reap the benefit."
Slavery.
Franklin's views and practices concerning slavery evolved over the course of his life. In his early years, Franklin owned seven slaves, including two men who worked in his household and his shop, but in his later years became an adherent of abolition. A revenue stream for his newspaper was paid ads for the sale of slaves and for the capture of runaway slaves and Franklin allowed the sale of slaves in his general store. He later became an outspoken critic of slavery. In 1758, he advocated the opening of a school for the education of black slaves in Philadelphia. He took two slaves to England with him, Peter and King. King escaped with a woman to live in the outskirts of London, and by 1758 he was working for a household in Suffolk. After returning from England in 1762, Franklin became more abolitionist in nature, attacking American slavery. In the wake of "Somerset v Stewart", he voiced frustration at British abolitionists:
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Franklin refused to publicly debate the issue of slavery at the 1787 Constitutional Convention.
At the time of the American founding, there were about half a million slaves in the United States, mostly in the five southernmost states, where they made up 40% of the population. Many of the leading American founderssuch as Thomas Jefferson, George Washington, and James Madisonowned slaves, but many others did not. Benjamin Franklin thought that slavery was "an atrocious debasement of human nature" and "a source of serious evils." In 1787, Franklin and Benjamin Rush helped write a new constitution for the Pennsylvania Society for Promoting the Abolition of Slavery, and that same year Franklin became president of the organization. In 1790, Quakers from New York and Pennsylvania presented their petition for abolition to Congress. Their argument against slavery was backed by the Pennsylvania Abolitionist Society.
In his later years, as Congress was forced to deal with the issue of slavery, Franklin wrote several essays that stressed the importance of the abolition of slavery and of the integration of African Americans into American society. These writings included:
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Vegetarianism.
Franklin became a vegetarian when he was a teenager apprenticing at a print shop, after coming upon a book by the early vegetarian advocate Thomas Tryon. In addition, he would have also been familiar with the moral arguments espoused by prominent vegetarian Quakers in the colonial-era Province of Pennsylvania, including Benjamin Lay and John Woolman. His reasons for vegetarianism were based on health, ethics, and economy:
Franklin also declared the consumption of fish to be "unprovoked murder." Despite his convictions, he began to eat fish after being tempted by fried cod on a boat sailing from Boston, justifying the eating of animals by observing that the fish's stomach contained other fish. Nonetheless, he recognized the faulty ethics in this argument and would continue to be a vegetarian on and off. He was "excited" by tofu, which he learned of from the writings of a Spanish missionary to Southeast Asia, Domingo Fernández Navarrete. Franklin sent a sample of soybeans to prominent American botanist John Bartram and had previously written to British diplomat and Chinese trade expert James Flint inquiring as to how tofu was made, with their correspondence believed to be the first documented use of the word "tofu" in the English language.
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Franklin's "Second Reply to "Vindex Patriae,"" a 1766 letter advocating self-sufficiency and less dependence on England, lists various examples of the bounty of American agricultural products, and does not mention meat. Detailing new American customs, he wrote that, "[t]hey resolved last spring to eat no more lamb; and not a joint of lamb has since been seen on any of their tables ... the sweet little creatures are all alive to this day, with the prettiest fleeces on their backs imaginable."
View on inoculation.
The concept of preventing smallpox by variolation was introduced to colonial America by an African slave named Onesimus via his owner Cotton Mather in the early eighteenth century, but the procedure was not immediately accepted. James Franklin's newspaper carried articles in 1721 that vigorously denounced the concept.
However, by 1736 Benjamin Franklin was known as a supporter of the procedure. Therefore, when four-year-old "Franky" died of smallpox, opponents of the procedure circulated rumors that the child had been inoculated, and that this was the cause of his subsequent death. When Franklin became aware of this gossip, he placed a notice in the "Pennsylvania Gazette", stating: "I do hereby sincerely declare, that he was not inoculated, but receiv'd the Distemper in the common Way of Infection ... I intended to have my Child inoculated." The child had a bad case of flux diarrhea, and his parents had waited for him to get well before having him inoculated. Franklin wrote in his "Autobiography": "In 1736 I lost one of my sons, a fine boy of four years old, by the small-pox, taken in the common way. I long regretted bitterly, and still regret that I had not given it to him by inoculation. This I mention for the sake of parents who omit that operation, on the supposition that they should never forgive themselves if a child died under it; my example showing that the regret may be the same either way, and that, therefore, the safer should be chosen."
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Interests and activities.
Musical endeavors.
Franklin is known to have played the violin, the harp, and the guitar. He also composed music, which included a string quartet in early classical style. While he was in London, he developed a much-improved version of the glass harmonica, in which the glasses rotate on a shaft, with the player's fingers held steady, instead of the other way around. He worked with the London glassblower Charles James to create it, and instruments based on his mechanical version soon found their way to other parts of Europe. Joseph Haydn, a fan of Franklin's enlightened ideas, had a glass harmonica in his instrument collection. Wolfgang Amadeus Mozart composed for Franklin's glass harmonica, as did Beethoven. Gaetano Donizetti used the instrument in the accompaniment to Amelia's aria "Par che mi dica ancora" in the tragic opera "Il castello di Kenilworth" (1821), as did Camille Saint-Saëns in his 1886 "The Carnival of the Animals". Richard Strauss calls for the glass harmonica in his 1917 "Die Frau ohne Schatten", and numerous other composers used Franklin's instrument as well.
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Chess.
Franklin was an avid chess player. He was playing chess by around 1733, making him the first chess player known by name in the American colonies. His essay on "The Morals of Chess" in "Columbian Magazine" in December 1786 is the second known writing on chess in America. This essay in praise of chess and prescribing a code of behavior for the game has been widely reprinted and translated. He and a friend used chess as a means of learning the Italian language, which both were studying; the winner of each game between them had the right to assign a task, such as parts of the Italian grammar to be learned by heart, to be performed by the loser before their next meeting.
Franklin was able to play chess more frequently against stronger opposition during his many years as a civil servant and diplomat in England, where the game was far better established than in America. He was able to improve his playing standard by facing more experienced players during this period. He regularly attended Old Slaughter's Coffee House in London for chess and socializing, making many important personal contacts. While in Paris, both as a visitor and later as ambassador, he visited the famous Café de la Régence, which France's strongest players made their regular meeting place. No records of his games have survived, so it is not possible to ascertain his playing strength in modern terms.
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Franklin was inducted into the U.S. Chess Hall of Fame in 1999. The Franklin Mercantile Chess Club in Philadelphia, the second oldest chess club in the U.S., is named in his honor.
Legacy.
Bequest.
Franklin bequeathed £1,000 (about $4,400 at the time, or about $125,000 in 2021 dollars) each to the cities of Boston and Philadelphia, in trust to gather interest for 200 years. The trust began in 1785 when the French mathematician Charles-Joseph Mathon de la Cour, who admired Franklin greatly, wrote a friendly parody of Franklin's "Poor Richard's Almanack" called "Fortunate Richard". The main character leaves a smallish amount of money in his will, five lots of 100 "livres", to collect interest over one, two, three, four or five full centuries, with the resulting astronomical sums to be spent on impossibly elaborate utopian projects. Franklin, who was 79 years old at the time, wrote thanking him for a great idea and telling him that he had decided to leave a bequest of 1,000 pounds each to his native Boston and his adopted Philadelphia.
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By 1990, more than $2,000,000 (~$ in ) had accumulated in Franklin's Philadelphia trust, which had loaned the money to local residents. From 1940 to 1990, the money was used mostly for mortgage loans. When the trust came due, Philadelphia decided to spend it on scholarships for local high school students. Franklin's Boston trust fund accumulated almost $5,000,000 during that same time; at the end of its first 100 years a portion was allocated to help establish a trade school that became the Franklin Institute of Boston, and the entire fund was later dedicated to supporting this institute.
In 1787, a group of prominent ministers in Lancaster, Pennsylvania, proposed the foundation of a new college named in Franklin's honor. Franklin donated £200 towards the development of Franklin College (now called Franklin & Marshall College).
Likeness and image.
As the only person to have signed the Declaration of Independence in 1776, Treaty of Alliance with France in 1778, Treaty of Paris in 1783, and U.S. Constitution in 1787, Franklin is considered one of the leading Founding Fathers of the United States. His pervasive influence in the early history of the nation has led to his being jocularly called "the only president of the United States who was never president of the United States."
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Franklin's likeness is ubiquitous. Since 1914, it has adorned American $100 bills. From 1948 to 1963, Franklin's portrait was on the half-dollar. He has appeared on a $50 bill and on several varieties of the $100 bill from 1914 and 1918. Franklin also appears on the $1,000 Series EE savings bond.
On April 12, 1976, as part of a bicentennial celebration, Congress dedicated a tall marble statue in Philadelphia's Franklin Institute as the Benjamin Franklin National Memorial. Vice President Nelson Rockefeller presided over the dedication ceremony. Many of Franklin's personal possessions are on display at the institute. In London, his house at 36 Craven Street, which is the only surviving former residence of Franklin, was first marked with a blue plaque and has since been opened to the public as the Benjamin Franklin House. In 1998, workmen restoring the building dug up the remains of six children and four adults hidden below the home. A total of 15 bodies have been recovered. The Friends of Benjamin Franklin House (the organization responsible for the restoration) note that the bones were likely placed there by William Hewson, who lived in the house for two years and who had built a small anatomy school at the back of the house. They note that while Franklin likely knew what Hewson was doing, he probably did not participate in any dissections because he was much more of a physicist than a medical man.
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He has been honored on U.S. postage stamps many times. The image of Franklin, the first postmaster general of the United States, occurs on the face of U.S. postage more than any other American save that of George Washington. He appeared on the first U.S. postage stamp issued in 1847. From 1908 through 1923, the U.S. Post Office issued a series of postage stamps commonly referred to as the Washington–Franklin Issues, in which Washington and Franklin were depicted many times over a 14-year period, the longest run of any one series in U.S. postal history. However, he only appears on a few . Some of the finest portrayals of Franklin on record can be found on the engravings inscribed on the face of U.S. postage. |
Banach space
In mathematics, more specifically 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 that is 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.
Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space".
Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces.
Definition.
A Banach space is a complete normed space formula_1
A normed space is a pair
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formula_2 consisting of a vector space formula_3 over a scalar field formula_4 (where formula_4 is commonly formula_6 or formula_7) together with a distinguished
norm formula_8 Like all norms, this norm induces a translation invariant
distance function, called the "canonical" or "(norm) induced metric", defined for all vectors formula_9 by
formula_10
This makes formula_3 into a metric space formula_12
A sequence formula_13 is called or or if for every real formula_14 there exists some index formula_15 such that
formula_16
whenever formula_17 and formula_18 are greater than formula_19
The normed space formula_2 is called a Banach space and the canonical metric formula_21 is called a "complete metric" if formula_22 is a complete metric space, which by definition means for every Cauchy sequence formula_13 in formula_24 there exists some formula_25 such that
formula_26
where because formula_27 this sequence's convergence to formula_28 can equivalently be expressed as
formula_29
The norm formula_30 of a normed space formula_2 is called a if formula_2 is a Banach space.
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L-semi-inner product.
For any normed space formula_33 there exists an L-semi-inner product formula_34 on formula_3 such that formula_36 for all formula_37 In general, there may be infinitely many L-semi-inner products that satisfy this condition and the proof of the existence of L-semi-inner products relies on the non-constructive Hahn–Banach theorem. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish Hilbert spaces from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces.
Characterization in terms of series.
The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors.
A normed space formula_3 is a Banach space if and only if each absolutely convergent series in formula_3 converges to a value that lies within formula_40 symbolically
formula_41
Topology.
The canonical metric formula_21 of a normed space formula_2 induces the usual metric topology formula_44 on formula_40 which is referred to as the "canonical" or "norm induced topology".
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Every normed space is automatically assumed to carry this Hausdorff topology, unless indicated otherwise.
With this topology, every Banach space is a Baire space, although there exist normed spaces that are Baire but not Banach. The norm formula_46 is always a continuous function with respect to the topology that it induces.
The open and closed balls of radius formula_47 centered at a point formula_25 are, respectively, the sets
formula_49
Any such ball is a convex and bounded subset of formula_40 but a compact ball/neighborhood exists if and only if formula_3 is finite-dimensional.
In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property.
If formula_52 is a vector and formula_53 is a scalar, then
formula_54
Using formula_55 shows that the norm-induced topology is translation invariant, which means that for any formula_25 and formula_57 the subset formula_58 is open (respectively, closed) in formula_3 if and only if its translation formula_60 is open (respectively, closed).
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Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include
formula_61
where formula_62 can be any sequence of positive real numbers that converges to formula_63 in formula_64 (common choices are formula_65 or formula_66).
So, for example, any open subset formula_67 of formula_3 can be written as a union
formula_69
indexed by some subset formula_70 where each formula_71 may be chosen from the aforementioned sequence formula_72 (The open balls can also be replaced with closed balls, although the indexing set formula_73 and radii formula_71 may then also need to be replaced).
Additionally, formula_73 can always be chosen to be countable if formula_3 is a , which by definition means that formula_3 contains some countable dense subset.
Homeomorphism classes of separable Banach spaces.
All finite–dimensional normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic.
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Every separable infinite–dimensional Hilbert space is linearly isometrically isomorphic to the separable Hilbert sequence space formula_78 with its usual norm formula_79
The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space formula_80 of countably many copies of formula_6 (this homeomorphism need not be a linear map).
Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other (or said differently, their topology is unique up to a homeomorphism).
Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including formula_82
In fact, formula_78 is even homeomorphic to its own unit formula_84 which stands in sharp contrast to finite–dimensional spaces (the Euclidean plane formula_85 is not homeomorphic to the unit circle, for instance).
This pattern in homeomorphism classes extends to generalizations of metrizable (locally Euclidean) topological manifolds known as , which are metric spaces that are around every point, locally homeomorphic to some open subset of a given Banach space (metric Hilbert manifolds and metric Fréchet manifolds are defined similarly).
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For example, every open subset formula_67 of a Banach space formula_3 is canonically a metric Banach manifold modeled on formula_3 since the inclusion map formula_89 is an open local homeomorphism.
Using Hilbert space microbundles, David Henderson showed in 1969 that every metric manifold modeled on a separable infinite–dimensional Banach (or Fréchet) space can be topologically embedded as an subset of formula_78 and, consequently, also admits a unique smooth structure making it into a formula_91 Hilbert manifold.
Compact and convex subsets.
There is a compact subset formula_58 of formula_78 whose convex hull formula_94 is closed and thus also compact.
However, like in all Banach spaces, the convex hull formula_95 of this (and every other) compact subset will be compact. In a normed space that is not complete then it is in general guaranteed that formula_95 will be compact whenever formula_58 is; an example can even be found in a (non-complete) pre-Hilbert vector subspace of formula_82
As a topological vector space.
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This norm-induced topology also makes formula_99 into what is known as a topological vector space (TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS formula_99 is a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is associated with particular norm or metric (both of which are "forgotten"). This Hausdorff TVS formula_99 is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex balanced open sets. This TVS is also , which by definition refers to any TVS whose topology is induced by some (possibly unknown) norm. Normable TVSs are characterized by being Hausdorff and having a bounded convex neighborhood of the origin.
All Banach spaces are barrelled spaces, which means that every barrel is neighborhood of the origin (all closed balls centered at the origin are barrels, for example) and guarantees that the Banach–Steinhaus theorem holds.
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Comparison of complete metrizable vector topologies.
The open mapping theorem implies that when formula_102 and formula_103 are topologies on formula_3 that make both formula_105 and formula_106 into complete metrizable TVSes (for example, Banach or Fréchet spaces), if one topology is finer or coarser than the other, then they must be equal (that is, if formula_107 or formula_108 then formula_109).
So, for example, if formula_110 and formula_111 are Banach spaces with topologies formula_112 and formula_113 and if one of these spaces has some open ball that is also an open subset of the other space (or, equivalently, if one of formula_114 or formula_115 is continuous), then their topologies are identical and the norms formula_116 and formula_117 are equivalent.
Completeness.
Complete norms and equivalent norms.
Two norms, formula_116 and formula_119 on a vector space formula_3 are said to be "equivalent" if they induce the same topology; this happens if and only if there exist real numbers formula_121 such that formula_122 for all formula_37 If formula_116 and formula_117 are two equivalent norms on a vector space formula_3 then formula_110 is a Banach space if and only if formula_111 is a Banach space.
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See this footnote for an example of a continuous norm on a Banach space that is equivalent to that Banach space's given norm.
All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.
Complete norms vs complete metrics.
A metric formula_129 on a vector space formula_3 is induced by a norm on formula_3 if and only if formula_129 is translation invariant and "absolutely homogeneous", which means that formula_133 for all scalars formula_134 and all formula_135 in which case the function formula_136 defines a norm on formula_3 and the canonical metric induced by formula_30 is equal to formula_139
Suppose that formula_2 is a normed space and that formula_141 is the norm topology induced on formula_142 Suppose that formula_129 is metric on formula_3 such that the topology that formula_129 induces on formula_3 is equal to formula_147 If formula_129 is translation invariant then formula_2 is a Banach space if and only if formula_150 is a complete metric space.
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If formula_129 is translation invariant, then it may be possible for formula_2 to be a Banach space but for formula_150 to be a complete metric space (see this footnote for an example). In contrast, a theorem of Klee, which also applies to all metrizable topological vector spaces, implies that if there exists complete metric formula_129 on formula_3 that induces the norm topology formula_141 on formula_40 then formula_2 is a Banach space.
A Fréchet space is a locally convex topological vector space whose topology is induced by some translation-invariant complete metric.
Every Banach space is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function (such as the space of real sequences formula_159 with the product topology).
However, the topology of every Fréchet space is induced by some countable family of real-valued (necessarily continuous) maps called seminorms, which are generalizations of norms.
It is even possible for a Fréchet space to have a topology that is induced by a countable family of (such norms would necessarily be continuous)
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but to not be a Banach/normable space because its topology can not be defined by any norm.
An example of such a space is the Fréchet space formula_160 whose definition can be found in the article on spaces of test functions and distributions.
Complete norms vs complete topological vector spaces.
There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space (TVS) or TVS-completeness, which uses the theory of uniform spaces.
Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity, called the canonical uniformity, that depends on vector subtraction and the topology formula_141 that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology formula_141 (and even applies to TVSs that are even metrizable).
Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space.
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If formula_163 is a metrizable topological vector space (such as any norm induced topology, for example), then formula_163 is a complete TVS if and only if it is a complete TVS, meaning that it is enough to check that every Cauchy in formula_163 converges in formula_163 to some point of formula_3 (that is, there is no need to consider the more general notion of arbitrary Cauchy nets).
If formula_163 is a topological vector space whose topology is induced by (possibly unknown) norm (such spaces are called ), then formula_163 is a complete topological vector space if and only if formula_3 may be assigned a norm formula_30 that induces on formula_3 the topology formula_141 and also makes formula_2 into a Banach space.
A Hausdorff locally convex topological vector space formula_3 is normable if and only if its strong dual space formula_176 is normable, in which case formula_176 is a Banach space (formula_176 denotes the strong dual space of formula_40 whose topology is a generalization of the dual norm-induced topology on the continuous dual space formula_180; see this footnote for more details).
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If formula_3 is a metrizable locally convex TVS, then formula_3 is normable if and only if formula_176 is a Fréchet–Urysohn space.
This shows that in the category of locally convex TVSs, Banach spaces are exactly those complete spaces that are both metrizable and have metrizable strong dual spaces.
Completions.
Every normed space can be isometrically embedded onto a dense vector subspace of a Banach space, where this Banach space is called a "completion" of the normed space. This Hausdorff completion is unique up to isometric isomorphism.
More precisely, for every normed space formula_40 there exists a Banach space formula_185 and a mapping formula_186 such that formula_187 is an isometric mapping and formula_188 is dense in formula_189 If formula_190 is another Banach space such that there is an isometric isomorphism from formula_3 onto a dense subset of formula_192 then formula_190 is isometrically isomorphic to formula_189
The Banach space formula_185 is the Hausdorff "completion" of the normed space formula_142 The underlying metric space for formula_185 is the same as the metric completion of formula_40 with the vector space operations extended from formula_3 to formula_189 The completion of formula_3 is sometimes denoted by formula_202
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General theory.
Linear operators, isomorphisms.
If formula_3 and formula_185 are normed spaces over the same ground field formula_205 the set of all continuous formula_4-linear maps formula_186 is denoted by formula_208 In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space formula_3 to another normed space is continuous if and only if it is bounded on the closed unit ball of formula_142 Thus, the vector space formula_211 can be given the operator norm
formula_212
For formula_185 a Banach space, the space formula_211 is a Banach space with respect to this norm. In categorical contexts, it is sometimes convenient to restrict the function space between two Banach spaces to only the short maps; in that case the space formula_215 reappears as a natural bifunctor.
If formula_3 is a Banach space, the space formula_217 forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.
If formula_3 and formula_185 are normed spaces, they are isomorphic normed spaces if there exists a linear bijection formula_186 such that formula_187 and its inverse formula_222 are continuous. If one of the two spaces formula_3 or formula_185 is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces formula_3 and formula_185 are "isometrically isomorphic" if in addition, formula_187 is an isometry, that is, formula_228 for every formula_28 in formula_142 The Banach–Mazur distance formula_231 between two isomorphic but not isometric spaces formula_3 and formula_185 gives a measure of how much the two spaces formula_3 and formula_185 differ.
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Continuous and bounded linear functions and seminorms.
Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a continuous function. So in particular, because the scalar field (which is formula_64 or formula_7) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.
If formula_238 is a subadditive function (such as a norm, a sublinear function, or real linear functional), then formula_239 is continuous at the origin if and only if formula_239 is uniformly continuous on all of formula_3; and if in addition formula_242 then formula_239 is continuous if and only if its absolute value formula_244 is continuous, which happens if and only if formula_245 is an open subset of formula_142
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And very importantly for applying the Hahn–Banach theorem, a linear functional formula_239 is continuous if and only if this is true of its real part formula_248 and moreover, formula_249 and the real part formula_248 completely determines formula_251 which is why the Hahn–Banach theorem is often stated only for real linear functionals.
Also, a linear functional formula_239 on formula_3 is continuous if and only if the seminorm formula_254 is continuous, which happens if and only if there exists a continuous seminorm formula_255 such that formula_256; this last statement involving the linear functional formula_239 and seminorm formula_116 is encountered in many versions of the Hahn–Banach theorem.
Basic notions.
The Cartesian product formula_259 of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, such as
formula_260
which correspond (respectively) to the coproduct and product in the category of Banach spaces and short maps (discussed above). For finite (co)products, these norms give rise to isomorphic normed spaces, and the product formula_259 (or the direct sum formula_262) is complete if and only if the two factors are complete.
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If formula_263 is a closed linear subspace of a normed space formula_40 there is a natural norm on the quotient space formula_265
formula_266
The quotient formula_267 is a Banach space when formula_3 is complete. The quotient map from formula_3 onto formula_265 sending formula_25 to its class formula_272 is linear, onto, and of norm formula_273 except when formula_274 in which case the quotient is the null space.
The closed linear subspace formula_263 of formula_3 is said to be a "complemented subspace" of formula_3 if formula_263 is the range of a surjective bounded linear projection formula_279 In this case, the space formula_3 is isomorphic to the direct sum of formula_263 and formula_282 the kernel of the projection formula_283
Suppose that formula_3 and formula_185 are Banach spaces and that formula_286 There exists a canonical factorization of formula_187 as
formula_288
where the first map formula_289 is the quotient map, and the second map formula_290 sends every class formula_291 in the quotient to the image formula_292 in formula_189 This is well defined because all elements in the same class have the same image. The mapping formula_290 is a linear bijection from formula_295 onto the range formula_296 whose inverse need not be bounded.
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Classical spaces.
Basic examples of Banach spaces include: the Lp spaces formula_297 and their special cases, the sequence spaces formula_298 that consist of scalar sequences indexed by natural numbers formula_299; among them, the space formula_300 of absolutely summable sequences and the space formula_301 of square summable sequences; the space formula_302 of sequences tending to zero and the space formula_303 of bounded sequences; the space formula_304 of continuous scalar functions on a compact Hausdorff space formula_305 equipped with the max norm,
formula_306
According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some formula_307 For every separable Banach space formula_40 there is a closed subspace formula_263 of formula_300 such that formula_311
Any Hilbert space serves as an example of a Banach space. A Hilbert space formula_312 on formula_313 is complete for a norm of the form
formula_314
where
formula_315
is the inner product, linear in its first argument that satisfies the following:
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formula_316
For example, the space formula_317 is a Hilbert space.
The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to formula_297 spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others.
Banach algebras.
A "Banach algebra" is a Banach space formula_319 over formula_320 or formula_321 together with a structure of algebra over formula_4, such that the product map formula_323 is continuous. An equivalent norm on formula_319 can be found so that formula_325 for all formula_326
Dual space.
If formula_3 is a normed space and formula_4 the underlying field (either the reals or the complex numbers), the "continuous dual space" is the space of continuous linear maps from formula_3 into formula_205 or "continuous linear functionals".
The notation for the continuous dual is formula_355 in this article.
Since formula_4 is a Banach space (using the absolute value as norm), the dual formula_180 is a Banach space, for every normed space formula_142 The Dixmier–Ng theorem characterizes the dual spaces of Banach spaces.
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The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.
In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional.
An important special case is the following: for every vector formula_28 in a normed space formula_40 there exists a continuous linear functional formula_239 on formula_3 such that
formula_363
When formula_28 is not equal to the formula_365 vector, the functional formula_239 must have norm one, and is called a "norming functional" for formula_367
The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane.
The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane.
A subset formula_58 in a Banach space formula_3 is "total" if the linear span of formula_58 is dense in formula_142 The subset formula_58 is total in formula_3 if and only if the only continuous linear functional that vanishes on formula_58 is the formula_365 functional: this equivalence follows from the Hahn–Banach theorem.
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If formula_3 is the direct sum of two closed linear subspaces formula_263 and formula_378 then the dual formula_180 of formula_3 is isomorphic to the direct sum of the duals of formula_263 and formula_19
If formula_263 is a closed linear subspace in formula_40 one can associate the formula_263 in the dual,
formula_386
The orthogonal formula_387 is a closed linear subspace of the dual. The dual of formula_263 is isometrically isomorphic to formula_389
The dual of formula_267 is isometrically isomorphic to formula_391
The dual of a separable Banach space need not be separable, but:
When formula_180 is separable, the above criterion for totality can be used for proving the existence of a countable total subset in formula_142
Weak topologies.
The "weak topology" on a Banach space formula_3 is the coarsest topology on formula_3 for which all elements formula_396 in the continuous dual space formula_180 are continuous.
The norm topology is therefore finer than the weak topology.
It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed.
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A norm-continuous linear map between two Banach spaces formula_3 and formula_185 is also "weakly continuous", that is, continuous from the weak topology of formula_3 to that of formula_189
If formula_3 is infinite-dimensional, there exist linear maps which are not continuous. The space formula_403 of all linear maps from formula_3 to the underlying field formula_4 (this space formula_403 is called the algebraic dual space, to distinguish it from formula_180 also induces a topology on formula_3 which is finer than the weak topology, and much less used in functional analysis.
On a dual space formula_409 there is a topology weaker than the weak topology of formula_409 called the "weak* topology".
It is the coarsest topology on formula_180 for which all evaluation maps formula_412 where formula_28 ranges over formula_40 are continuous.
Its importance comes from the Banach–Alaoglu theorem.
The Banach–Alaoglu theorem can be proved using Tychonoff's theorem about infinite products of compact Hausdorff spaces.
When formula_3 is separable, the unit ball formula_416 of the dual is a metrizable compact in the weak* topology.
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Examples of dual spaces.
The dual of formula_302 is isometrically isomorphic to formula_300: for every bounded linear functional formula_239 on formula_420 there is a unique element formula_421 such that
formula_422
The dual of formula_300 is isometrically isomorphic to formula_303.
The dual of Lebesgue space formula_425 is isometrically isomorphic to formula_426 when formula_427 and formula_428
For every vector formula_429 in a Hilbert space formula_430 the mapping
formula_431
defines a continuous linear functional formula_432 on formula_433The Riesz representation theorem states that every continuous linear functional on formula_312 is of the form formula_432 for a uniquely defined vector formula_429 in formula_433
The mapping formula_438 is an antilinear isometric bijection from formula_312 onto its dual formula_440
When the scalars are real, this map is an isometric isomorphism.
When formula_348 is a compact Hausdorff topological space, the dual formula_442 of formula_304 is the space of Radon measures in the sense of Bourbaki.
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The subset formula_444 of formula_442 consisting of non-negative measures of mass 1 (probability measures) is a convex w*-closed subset of the unit ball of formula_446
The extreme points of formula_444 are the Dirac measures on formula_448
The set of Dirac measures on formula_305 equipped with the w*-topology, is homeomorphic to formula_448
The result has been extended by Amir and Cambern to the case when the multiplicative Banach–Mazur distance between formula_304 and formula_452 is formula_453
The theorem is no longer true when the distance is formula_454
In the commutative Banach algebra formula_455 the maximal ideals are precisely kernels of Dirac measures on formula_305
formula_457
More generally, by the Gelfand–Mazur theorem, the maximal ideals of a unital commutative Banach algebra can be identified with its characters—not merely as sets but as topological spaces: the former with the hull-kernel topology and the latter with the w*-topology.
In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual formula_458
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Not every unital commutative Banach algebra is of the form formula_304 for some compact Hausdorff space formula_448 However, this statement holds if one places formula_304 in the smaller category of commutative C*-algebras.
Gelfand's representation theorem for commutative C*-algebras states that every commutative unital "C"*-algebra formula_319 is isometrically isomorphic to a formula_304 space.
The Hausdorff compact space formula_348 here is again the maximal ideal space, also called the spectrum of formula_319 in the C*-algebra context.
Bidual.
If formula_3 is a normed space, the (continuous) dual formula_467 of the dual formula_180 is called the ' or ' of formula_142
For every normed space formula_40 there is a natural map,
formula_471
This defines formula_472 as a continuous linear functional on formula_409 that is, an element of formula_474 The map formula_475 is a linear map from formula_3 to formula_474
As a consequence of the existence of a norming functional formula_239 for every formula_479 this map formula_480 is isometric, thus injective.
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For example, the dual of formula_481 is identified with formula_482 and the dual of formula_300 is identified with formula_484 the space of bounded scalar sequences.
Under these identifications, formula_480 is the inclusion map from formula_302 to formula_487 It is indeed isometric, but not onto.
If formula_480 is surjective, then the normed space formula_3 is called "reflexive" (see below).
Being the dual of a normed space, the bidual formula_467 is complete, therefore, every reflexive normed space is a Banach space.
Using the isometric embedding formula_491 it is customary to consider a normed space formula_3 as a subset of its bidual.
When formula_3 is a Banach space, it is viewed as a closed linear subspace of formula_474 If formula_3 is not reflexive, the unit ball of formula_3 is a proper subset of the unit ball of formula_474
The Goldstine theorem states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual.
In other words, for every formula_498 in the bidual, there exists a net formula_499 in formula_3 so that
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formula_501
The net may be replaced by a weakly*-convergent sequence when the dual formula_180 is separable.
On the other hand, elements of the bidual of formula_300 that are not in formula_300 cannot be weak*-limit of in formula_482 since formula_300 is weakly sequentially complete.
Banach's theorems.
Here are the main general results about Banach spaces that go back to the time of Banach's book () and are related to the Baire category theorem.
According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors.
Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.
The Banach–Steinhaus theorem is not limited to Banach spaces.
It can be extended for example to the case where formula_3 is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood formula_67 of formula_365 in formula_3 such that all formula_187 in formula_512 are uniformly bounded on formula_513
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formula_514
This result is a direct consequence of the preceding "Banach isomorphism theorem" and of the canonical factorization of bounded linear maps.
This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from formula_515 onto formula_3 sending formula_517 to the sum formula_518
Reflexivity.
The normed space formula_3 is called "reflexive" when the natural map
formula_520
is surjective. Reflexive normed spaces are Banach spaces.
This is a consequence of the Hahn–Banach theorem.
Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space formula_3 onto the Banach space formula_522 then formula_185 is reflexive.
Indeed, if the dual formula_524 of a Banach space formula_185 is separable, then formula_185 is separable.
If formula_3 is reflexive and separable, then the dual of formula_180 is separable, so formula_180 is separable.
Hilbert spaces are reflexive. The formula_297 spaces are reflexive when formula_531 More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem.
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The spaces formula_532 are not reflexive.
In these examples of non-reflexive spaces formula_40 the bidual formula_467 is "much larger" than formula_142
Namely, under the natural isometric embedding of formula_3 into formula_467 given by the Hahn–Banach theorem, the quotient formula_538 is infinite-dimensional, and even nonseparable.
However, Robert C. James has constructed an example of a non-reflexive space, usually called "the James space" and denoted by formula_539 such that the quotient formula_540 is one-dimensional.
Furthermore, this space formula_541 is isometrically isomorphic to its bidual.
When formula_3 is reflexive, it follows that all closed and bounded convex subsets of formula_3 are weakly compact.
In a Hilbert space formula_430 the weak compactness of the unit ball is very often used in the following way: every bounded sequence in formula_312 has weakly convergent subsequences.
Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems.
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For example, every convex continuous function on the unit ball formula_546 of a reflexive space attains its minimum at some point in formula_547
As a special case of the preceding result, when formula_3 is a reflexive space over formula_549 every continuous linear functional formula_239 in formula_180 attains its maximum formula_552 on the unit ball of formula_142
The following theorem of Robert C. James provides a converse statement.
The theorem can be extended to give a characterization of weakly compact convex sets.
On every non-reflexive Banach space formula_40 there exist continuous linear functionals that are not "norm-attaining".
However, the Bishop–Phelps theorem states that norm-attaining functionals are norm dense in the dual formula_180 of formula_142
Weak convergences of sequences.
A sequence formula_557 in a Banach space formula_3 is "weakly convergent" to a vector formula_25 if formula_560 converges to formula_561 for every continuous linear functional formula_239 in the dual formula_563 The sequence formula_557 is a "weakly Cauchy sequence" if formula_560 converges to a scalar limit formula_566 for every formula_239 in formula_563
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A sequence formula_569 in the dual formula_180 is "weakly* convergent" to a functional formula_571 if formula_572 converges to formula_561 for every formula_28 in formula_142
Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.
When the sequence formula_557 in formula_3 is a weakly Cauchy sequence, the limit formula_578 above defines a bounded linear functional on the dual formula_409 that is, an element formula_578 of the bidual of formula_40 and formula_578 is the limit of formula_557 in the weak*-topology of the bidual.
The Banach space formula_3 is "weakly sequentially complete" if every weakly Cauchy sequence is weakly convergent in formula_142
It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.
An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the formula_365 vector.
The unit vector basis of formula_298 for formula_588 or of formula_420 is another example of a "weakly null sequence", that is, a sequence that converges weakly to formula_590
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For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to formula_590
The unit vector basis of formula_300 is not weakly Cauchy.
Weakly Cauchy sequences in formula_300 are weakly convergent, since formula_594-spaces are weakly sequentially complete.
Actually, weakly convergent sequences in formula_300 are norm convergent. This means that formula_300 satisfies Schur's property.
Results involving the basis.
Weakly Cauchy sequences and the formula_300 basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.
A complement to this result is due to Odell and Rosenthal (1975).
By the Goldstine theorem, every element of the unit ball formula_598 of formula_467 is weak*-limit of a net in the unit ball of formula_142 When formula_3 does not contain formula_482 every element of formula_598 is weak*-limit of a in the unit ball of formula_142
When the Banach space formula_3 is separable, the unit ball of the dual formula_409 equipped with the weak*-topology, is a metrizable compact space formula_305 and every element formula_498 in the bidual formula_467 defines a bounded function on formula_348:
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formula_611
This function is continuous for the compact topology of formula_348 if and only if formula_498 is actually in formula_40 considered as subset of formula_474
Assume in addition for the rest of the paragraph that formula_3 does not contain formula_617
By the preceding result of Odell and Rosenthal, the function formula_498 is the pointwise limit on formula_348 of a sequence formula_620 of continuous functions on formula_305 it is therefore a first Baire class function on formula_448
The unit ball of the bidual is a pointwise compact subset of the first Baire class on formula_448
Sequences, weak and weak* compactness.
When formula_3 is separable, the unit ball of the dual is weak*-compact by the Banach–Alaoglu theorem and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences.
This applies to separable reflexive spaces, but more is true in this case, as stated below.
The weak topology of a Banach space formula_3 is metrizable if and only if formula_3 is finite-dimensional. If the dual formula_180 is separable, the weak topology of the unit ball of formula_3 is metrizable.
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This applies in particular to separable reflexive Banach spaces.
Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.
A Banach space formula_3 is reflexive if and only if each bounded sequence in formula_3 has a weakly convergent subsequence.
A weakly compact subset formula_319 in formula_300 is norm-compact. Indeed, every sequence in formula_319 has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of formula_617
Type and cotype.
A way to classify Banach spaces is through the probabilistic notion of type and cotype, these two measure how far a Banach space is from a Hilbert space.
Schauder bases.
A "Schauder basis" in a Banach space formula_3 is a sequence formula_636 of vectors in formula_3 with the property that for every vector formula_479 there exist defined scalars formula_639 depending on formula_640 such that
formula_641
Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.
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It follows from the Banach–Steinhaus theorem that the linear mappings formula_642 are uniformly bounded by some constant formula_643
Let formula_644 denote the coordinate functionals which assign to every formula_28 in formula_3 the coordinate formula_647 of formula_28 in the above expansion.
They are called "biorthogonal functionals". When the basis vectors have norm formula_273 the coordinate functionals formula_644 have norm formula_651 in the dual of formula_142
Most classical separable spaces have explicit bases.
The Haar system formula_653 is a basis for formula_425 when formula_655
The trigonometric system is a basis in formula_656 when formula_531
The Schauder system is a basis in the space formula_658
The question of whether the disk algebra formula_328 has a basis remained open for more than forty years, until Bočkarev showed in 1974 that formula_328 admits a basis constructed from the Franklin system.
Since every vector formula_28 in a Banach space formula_3 with a basis is the limit of formula_663 with formula_664 of finite rank and uniformly bounded, the space formula_3 satisfies the bounded approximation property.
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The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.
Robert C. James characterized reflexivity in Banach spaces with a basis: the space formula_3 with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.
In this case, the biorthogonal functionals form a basis of the dual of formula_142
Tensor product.
Let formula_3 and formula_185 be two formula_4-vector spaces. The tensor product formula_671 of formula_3 and formula_185 is a formula_4-vector space formula_190 with a bilinear mapping formula_676 which has the following universal property:
The image under formula_187 of a couple formula_683 in formula_259 is denoted by formula_685 and called a "simple tensor".
Every element formula_686 in formula_671 is a finite sum of such simple tensors.
There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955.
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In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the "projective tensor product" of two Banach spaces formula_3 and formula_185 is the formula_690 of the algebraic tensor product formula_671 equipped with the projective tensor norm, and similarly for the "injective tensor product" formula_692
Grothendieck proved in particular that
formula_693
where formula_348 is a compact Hausdorff space, formula_695 the Banach space of continuous functions from formula_348 to formula_185 and formula_698 the space of Bochner-measurable and integrable functions from formula_699 to formula_522 and where the isomorphisms are isometric.
The two isomorphisms above are the respective extensions of the map sending the tensor formula_701 to the vector-valued function formula_702
Tensor products and the approximation property.
Let formula_3 be a Banach space. The tensor product formula_704 is identified isometrically with the closure in formula_339 of the set of finite rank operators.
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When formula_3 has the approximation property, this closure coincides with the space of compact operators on formula_142
For every Banach space formula_522 there is a natural norm formula_709 linear map
formula_710
obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem to the question of whether this map is one-to-one when formula_185 is the dual of formula_142
Precisely, for every Banach space formula_40 the map
formula_714
is one-to-one if and only if formula_3 has the approximation property.
Grothendieck conjectured that formula_690 and formula_717 must be different whenever formula_3 and formula_185 are infinite-dimensional Banach spaces.
This was disproved by Gilles Pisier in 1983.
Pisier constructed an infinite-dimensional Banach space formula_3 such that formula_721 and formula_722 are equal. Furthermore, just as Enflo's example, this space formula_3 is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space formula_724 does not have the approximation property.
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Some classification results.
Characterizations of Hilbert space among Banach spaces.
A necessary and sufficient condition for the norm of a Banach space formula_3 to be associated to an inner product is the parallelogram identity:
It follows, for example, that the Lebesgue space formula_425 is a Hilbert space only when formula_727
If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives:
formula_728
For complex scalars, defining the inner product so as to be formula_7-linear in formula_640 antilinear in formula_731 the polarization identity gives:
formula_732
To see that the parallelogram law is sufficient, one observes in the real case that formula_733 is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and formula_734 The parallelogram law implies that formula_733 is additive in formula_367
It follows that it is linear over the rationals, thus linear by continuity.
Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available.
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The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant formula_737: Kwapień proved that if
formula_738
for every integer formula_18 and all families of vectors formula_740 then the Banach space formula_3 is isomorphic to a Hilbert space.
Here, formula_742 denotes the average over the formula_743 possible choices of signs formula_744
In the same article, Kwapień proved that the validity of a Banach-valued Parseval's theorem for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces.
Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space. The proof rests upon Dvoretzky's theorem about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer formula_745 any finite-dimensional normed space, with dimension sufficiently large compared to formula_745 contains subspaces nearly isometric to the formula_18-dimensional Euclidean space.
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The next result gives the solution of the so-called . An infinite-dimensional Banach space formula_3 is said to be "homogeneous" if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to formula_301 is homogeneous, and Banach asked for the converse.
An infinite-dimensional Banach space is "hereditarily indecomposable" when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces.
The Gowers dichotomy theorem asserts that every infinite-dimensional Banach space formula_3 contains, either a subspace formula_185 with unconditional basis, or a hereditarily indecomposable subspace formula_192 and in particular, formula_190 is not isomorphic to its closed hyperplanes.
If formula_3 is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and Tomczak–Jaegermann, for spaces with an unconditional basis, that formula_3 is isomorphic to formula_756
Metric classification.
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If formula_186 is an isometry from the Banach space formula_3 onto the Banach space formula_185 (where both formula_3 and formula_185 are vector spaces over formula_64), then the Mazur–Ulam theorem states that formula_187 must be an affine transformation.
In particular, if formula_764 this is formula_187 maps the zero of formula_3 to the zero of formula_522 then formula_187 must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.
Topological classification.
Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces.
Anderson–Kadec theorem (1965–66) proves that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.
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Spaces of continuous functions.
When two compact Hausdorff spaces formula_769 and formula_770 are homeomorphic, the Banach spaces formula_771 and formula_772 are isometric. Conversely, when formula_769 is not homeomorphic to formula_774 the (multiplicative) Banach–Mazur distance between formula_771 and formula_772 must be greater than or equal to formula_777 see above the results by Amir and Cambern.
Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:
The situation is different for countably infinite compact Hausdorff spaces.
Every countably infinite compact formula_348 is homeomorphic to some closed interval of ordinal numbers
formula_779
equipped with the order topology, where formula_780 is a countably infinite ordinal.
The Banach space formula_304 is then isometric to . When formula_782 are two countably infinite ordinals, and assuming formula_783 the spaces and are isomorphic if and only if .
For example, the Banach spaces
formula_784
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are mutually non-isomorphic.
Derivatives.
Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gateaux derivative for details.
The Fréchet derivative allows for an extension of the concept of a total derivative to Banach spaces. The Gateaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces.
Fréchet differentiability is a stronger condition than Gateaux differentiability.
The quasi-derivative is another generalization of directional derivative that implies a stronger condition than Gateaux differentiability, but a weaker condition than Fréchet differentiability.
Generalizations.
Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions formula_785 or the space of all distributions on formula_549 are complete but are not normed vector spaces and hence not Banach spaces.
In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces. |
Bram Stoker
Abraham Stoker (8 November 1847 – 20 April 1912), better known by his pen name Bram Stoker, was an Irish novelist who wrote the 1897 Gothic horror novel "Dracula". The book is widely considered a milestone in Vampire fiction, and one of the most famous classics of English literature. The character of the primary antagonist Count Dracula ranks among the most iconic fictional figures of the entire Victorian era, and led to countless adaptations of the character for films, movies, plays, comics, video games, and stage performances.
During his life, he was better known as the personal assistant of the actor Sir Henry Irving and business manager of the West End's Lyceum Theatre, which Irving owned. Stoker was also a distant relative of Sir Arthur Conan Doyle, the creator of the fictional Sherlock Holmes crime detective character. The two novelists collaborated in writing other novels such as "The Fate of Fenella" in 1892.
In his early years, Stoker worked as a theatre critic for an Irish newspaper and wrote stories as well as commentaries. He also enjoyed travelling, particularly to Cruden Bay in Scotland where he set two of his novels and drew inspiration for writing "Dracula". He died on 20 April 1912 due to locomotor ataxia and was cremated in north London. Since his death, his magnum opus "Dracula" has become one of the best-selling works of vampire literature, and a classic of the genre.
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Early life.
Stoker was born on 8 November 1847 at 15 Marino Crescent, Clontarf, in Dublin, Ireland. The park adjacent to the house is now known as Bram Stoker Park. His parents were Abraham Stoker (1799–1876), an Anglo-Irishman from Dublin and Charlotte Mathilda Blake Thornley (1818–1901), of English and Irish descent, who was raised in County Sligo. Stoker was the third of seven children, the eldest of whom was Sir Thornley Stoker, 1st Baronet. Abraham and Charlotte were members of the Church of Ireland Parish of Clontarf and attended the parish church with their children, who were baptised there. Abraham was a senior civil servant.
Stoker was bedridden with an unknown illness until he started school at the age of seven when he made a complete recovery. Of this time, Stoker wrote, "I was naturally thoughtful, and the leisure of long illness gave opportunity for many thoughts which were fruitful according to their kind in later years." He was privately educated at Bective House school run by the Reverend William Woods.
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After his recovery, he grew up without further serious illnesses, even excelling as an athlete at Trinity College, Dublin, which he attended from 1864 to 1870. He graduated with a BA in 1870 and paid to receive his MA in 1875. Though he later in life recalled graduating "with honours in mathematics", this appears to have been a mistake. He was named University Athlete, participating in multiple sports, including playing rugby for Dublin University. He was auditor of the College Historical Society ("the Hist") and president of the University Philosophical Society (he remains the only student in Trinity's history to hold both positions), where his first paper was on "Sensationalism in Fiction and Society".
Early career.
Stoker became interested in the theatre while a student through his friend Dr. Maunsell. While working for the Irish Civil Service, he became the theatre critic for the "Dublin Evening Mail", which was co-owned by Sheridan Le Fanu, an author of Gothic tales. Theatre critics were held in low esteem at the time, but Stoker attracted notice by the quality of his reviews. In December 1876, he gave a favourable review of Henry Irving's "Hamlet" at the Theatre Royal in Dublin. Irving invited Stoker for dinner at the Shelbourne Hotel where he was staying, and they became friends. Stoker also wrote stories, and "Crystal Cup" was published by the London Society in 1872, followed by "The Chain of Destiny" in four parts in "The Shamrock". In 1876, while a civil servant in Dublin, Stoker wrote the non-fiction book "The Duties of Clerks of Petty Sessions in Ireland" (published 1879), which remained a standard work. Furthermore, he possessed an interest in art and was a founder of the Dublin Sketching Club in 1879.
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Lyceum Theatre.
In 1878, Stoker married Florence Balcombe, daughter of Lieutenant-Colonel James Balcombe of 1 Marino Crescent. She was a celebrated beauty whose former suitor had been Oscar Wilde. Stoker had known Wilde from his student days, having proposed him for membership of the university's Philosophical Society while he was president. Wilde was upset at Florence's decision, but Stoker later resumed the acquaintanceship, and, after Wilde's fall, visited him on the Continent.
The Stokers moved to London, where Stoker became acting manager and then business manager of Irving's Lyceum Theatre in the West End, a post he held for 27 years. On 31 December 1879, Bram and Florence's only child was born, a son whom they christened Irving Noel Thornley Stoker. The collaboration with Henry Irving was important for Stoker and through him, he became involved in London's high society, where he met James Abbott McNeill Whistler and Sir Arthur Conan Doyle (to whom he was distantly related). Working for Irving, the most famous actor of his time, and managing one of the most successful theatres in London made Stoker a notable if busy man. He was dedicated to Irving and his memoirs show he idolised him. In London, Stoker also met Hall Caine, who became one of his closest friends – he dedicated "Dracula" to him.
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In the course of Irving's tours, Stoker travelled the world, although he never visited Eastern Europe, a setting for his most famous novel. Stoker enjoyed the United States, where Irving was popular. With Irving, he was invited twice to the White House and knew William McKinley and Theodore Roosevelt. Stoker set two of his novels in America and used Americans as characters, the most notable being Quincey Morris. He also met one of his literary idols, Walt Whitman, having written to him in 1872 an extraordinary letter that some have interpreted as the expression of a deeply-suppressed homosexuality.
Bram Stoker in Cruden Bay.
Stoker was a regular visitor to Cruden Bay in Scotland between 1892 and 1910. His month-long holidays to the Aberdeenshire coastal village provided a large portion of available time for writing his books. Two novels were set in Cruden Bay: "The Watter's Mou' "(1895) and "The Mystery of the Sea" (1902). He started writing "Dracula" there in 1895 while in residence at the Kilmarnock Arms Hotel. The guest book with his signatures from 1894 and 1895 still survives. The nearby Slains Castle (also known as New Slains Castle) is linked with Bram Stoker and plausibly provided the visual palette for the descriptions of Castle Dracula during the writing phase. A distinctive room in Slains Castle, the octagonal hall, matches the description of the octagonal room in Castle Dracula.
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Writings.
Stoker visited the English coastal town of Whitby in 1890, and that visit was said to be part of the inspiration for "Dracula", staying at a guesthouse in West Cliff at 6 Royal Crescent, doing his research at the public library at 7 Pier Road (now "Quayside Fish and Chips"). Count Dracula comes ashore at Whitby, and in the shape of a black dog runs up the 199 steps to the graveyard of St Mary's Church in the shadow of the Whitby Abbey ruins. Stoker began writing novels while working as manager for Irving and secretary and director of London's Lyceum Theatre, beginning with "The Snake's Pass" in 1890 and "Dracula" in 1897. During this period, he was part of the literary staff of "The Daily Telegraph" in London, and he wrote other fiction, including the horror novels "The Lady of the Shroud" (1909) and "The Lair of the White Worm" (1911). He published his "Personal Reminiscences of Henry Irving" in 1906, after Irving's death, which proved successful, and managed productions at the Prince of Wales Theatre.
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Before writing "Dracula", Stoker met Ármin Vámbéry, a Hungarian-Jewish writer and traveller (born in Szent-György, Kingdom of Hungary now Svätý Jur, Slovakia). Dracula likely emerged from Vámbéry's dark stories of the Carpathian Mountains. However this claim has been challenged by many including Elizabeth Miller, a professor who, since 1990, has had as her major field of research and writing "Dracula", and its author, sources, and influences. She has stated, "The only comment about the subject matter of the talk was that Vambery 'spoke loudly against Russian aggression.'" There had been nothing in their conversations about the "tales of the terrible Dracula" that are supposed to have "inspired Stoker to equate his vampire-protagonist with the long-dead tyrant." At any rate, by this time, Stoker's novel was well underway, and he was already using the name Dracula for his vampire. Stoker then spent several years researching Central and East European folklore and mythological stories of vampires.
The 1972 book "In Search of Dracula" by Radu Florescu and Raymond McNally claimed that the Count in Stoker's novel was based on Vlad III Dracula. However, according to Elizabeth Miller, Stoker borrowed only the name and "scraps of miscellaneous information" about Romanian history; further, there are no comments about Vlad III in the author's working notes.
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"Dracula" is an epistolary novel, written as a collection of realistic but completely fictional diary entries, telegrams, letters, ship's logs, and newspaper clippings, all of which added a level of detailed realism to the story, a skill which Stoker had developed as a newspaper writer. At the time of its publication, "Dracula" was considered a "straightforward horror novel" based on imaginary creations of supernatural life. "It gave form to a universal fantasy ... and became a part of popular culture."
According to the "Encyclopedia of World Biography", Stoker's stories are today included in the categories of horror fiction, romanticized Gothic stories, and melodrama. They are classified alongside other works of popular fiction, such as Mary Shelley's "Frankenstein", which also used the myth-making and story-telling method of having multiple narrators telling the same tale from different perspectives. According to historian Jules Zanger, this leads the reader to the assumption that "they can't all be lying".
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The original 541-page typescript of "Dracula" was believed to have been lost until it was found in a barn in northwestern Pennsylvania in the early 1980s. It consisted of typed sheets with many emendations, and handwritten on the title page was "THE UN-DEAD." The author's name was shown at the bottom as Bram Stoker. Author Robert Latham remarked: "the most famous horror novel ever published, its title changed at the last minute". The typescript was purchased by Microsoft co-founder Paul Allen.
Stoker's inspirations for the story, in addition to Whitby, may have included a visit to Slains Castle in Aberdeenshire, a visit to the crypts of St. Michan's Church in Dublin, and the novella "Carmilla" by Sheridan Le Fanu.
Stoker's original research notes for the novel are kept by the Rosenbach Museum and Library in Philadelphia. A facsimile edition of the notes was created by Elizabeth Miller and Robert Eighteen-Bisang in 1998.
Stoker at the London Library.
Stoker was a member of the London Library and conducted much of the research for "Dracula" there. In 2018, the Library discovered some of the books that Stoker used for his research, complete with notes and marginalia.
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Death.
After suffering a number of strokes, Stoker died at No. 26 St George's Square, London on 20 April 1912. Some biographers attribute the cause of death to overwork, others to tertiary syphilis. His death certificate listed the cause of death as "Locomotor ataxia 6 months", presumed to be a reference to syphilis. He was cremated, and his ashes were placed in a display urn at Golders Green Crematorium in north London. The ashes of Irving Noel Stoker, the author's son, were added to his father's urn following his death in 1961. The original plan had been to keep his parents' ashes together, but after Florence Stoker's death, her ashes were scattered at the Gardens of Rest. His ashes are still stored in Golders Green Crematorium today.
Beliefs and philosophy.
Stoker was raised a Protestant in the Church of Ireland. He was a strong supporter of the Liberal Party and took a keen interest in Irish affairs. As a "philosophical home ruler", he supported Home Rule for Ireland brought about by peaceful means. He remained an ardent monarchist who believed that Ireland should remain within the British Empire. He was an admirer of Prime Minister William Ewart Gladstone, whom he knew personally, and supported his plans for Ireland.
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Stoker believed in progress and took a keen interest in science and science-based medicine. Some of Stoker's novels represent early examples of science fiction, such as "The Lady of the Shroud" (1909). He had a writer's interest in the occult, notably mesmerism, but despised fraud and believed in the superiority of the scientific method over superstition. Stoker counted among his friends J. W. Brodie-Innis, a member of the Hermetic Order of the Golden Dawn, and hired member Pamela Colman Smith as an artist for the Lyceum Theatre, but no evidence suggests that Stoker ever joined the Order himself.
Like Irving, who was an active Freemason, Stoker also became a member of the order, "initiated into Freemasonry in Buckingham and Chandos Lodge No. 1150 in February 1883, passed in April of that same year, and raised to the degree of Master Mason on 20 June 1883." Stoker however was not a particular active Freemason, spent only six years as an active member, and did not take part in any Masonic activities during his time in London.
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Posthumous.
The short story collection "Dracula's Guest and Other Weird Stories" was published in 1914 by Stoker's widow, Florence Stoker, who was also his literary executrix. The first film adaptation of "Dracula" was F. W. Murnau's "Nosferatu", released in 1922, with Max Schreck starring as Count Orlok. Florence Stoker eventually sued the filmmakers and was represented by the attorneys of the British Incorporated Society of Authors. Her chief legal complaint was that she had neither been asked for permission for the adaptation nor paid any royalty. The case dragged on for some years, with Mrs. Stoker demanding the destruction of the negative and all prints of the film. The suit was finally resolved in the widow's favour in July 1925. A single print of the film survived, however, and it has become well known. The first authorised film version of "Dracula" did not come about until almost a decade later when Universal Studios released Tod Browning's "Dracula" starring Bela Lugosi.
Dacre Stoker.
Canadian writer Dacre Stoker, a great-grandnephew of Bram Stoker, decided to write "a sequel that bore the Stoker name" to "reestablish creative control over" the original novel, with encouragement from screenwriter Ian Holt, because of the Stokers' frustrating history with "Dracula's" copyright. In 2009, "Dracula: The Un-Dead" was released, written by Dacre Stoker and Ian Holt. Both writers "based [their work] on Bram Stoker's own handwritten notes for characters and plot threads excised from the original edition" along with their own research for the sequel. This also marked Dacre Stoker's writing debut.
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In spring 2012, Dacre Stoker in collaboration with Elizabeth Miller presented the "lost" Dublin Journal written by Bram Stoker, which had been kept by his great-grandson Noel Dobbs. Stoker's diary entries shed a light on the issues that concerned him before his London years. A remark about a boy who caught flies in a bottle might be a clue for the later development of the Renfield character in "Dracula".
Commemorations.
On 8 November 2012, Stoker was honoured with a Google Doodle on Google's homepage commemorating the 165th anniversary of his birth.
An annual festival takes place in Dublin, the birthplace of Bram Stoker, in honour of his literary achievements. The Dublin City Council Bram Stoker Festival encompasses spectacles, literary events, film, family-friendly activities and outdoor events, and takes place every October Bank Holiday Weekend in Dublin. The festival is supported by the Bram Stoker Estate and is funded by Dublin City Council. |
Billion (disambiguation)
Billion is a name for a large number. It may refer specifically to:
Billion may also refer to: |
Contract bridge
Contract bridge, or simply bridge, is a trick-taking card game using a standard 52-card deck. In its basic format, it is played by four players in two competing partnerships, with partners sitting opposite each other around a table. Millions of people play bridge worldwide in clubs, tournaments, online and with friends at home, making it one of the world's most popular card games, particularly among seniors. The World Bridge Federation (WBF) is the governing body for international competitive bridge, with numerous other bodies governing it at the regional level.
The game consists of a number of , each progressing through four phases. The cards are to the players; then the players "call" (or "bid") in an seeking to take the , specifying how many tricks the partnership receiving the contract (the declaring side) needs to take to receive points for the deal. During the auction, partners use their bids to exchange information about their hands, including overall strength and distribution of the suits; no other means of conveying or implying any information is permitted. The cards are then played, the trying to fulfill the contract, and the trying to stop the declaring side from achieving its goal. The deal is scored based on the number of tricks taken, the contract, and various other factors which depend to some extent on the variation of the game being played.
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Rubber bridge is the most popular variation for casual play, but most club and tournament play involves some variant of duplicate bridge, where the cards are not re-dealt on each occasion, but the same deal is played by two or more sets of players (or "tables") to enable comparative scoring.
History and etymology.
Bridge is a member of the family of trick-taking games and is a derivative of whist, which had become the dominant such game and enjoyed a loyal following for centuries. The idea of a trick-taking, 52-card game has its first documented origins in Italy and France. The French physician and author Rabelais (1493–1553) mentions a game called "La Triomphe" in one of his works, and Juan Luis Vives's "Linguae latinae exercitio" (Exercise in the Latin language) of 1539 features a dialogue on card games in which the characters play 'Triumphus hispanicus' (Spanish Triumph).
Bridge departed from whist with the creation of "Biritch" in the 19th century and evolved through the late 19th and early 20th centuries to form the present game. The first known rule book for bridge, dated 1886, is "" written by John Collinson, an English financier working in Ottoman Constantinople. It and his subsequent letter to "The Saturday Review", dated 28 May 1906, document the origin of "Biritch" as being the Russian community in Constantinople. The word "biritch" is thought to be a transliteration of the Russian word (бирчий, бирич), an occupation of a diplomatic clerk or an announcer. Another theory is that British soldiers invented the game bridge while serving in the Crimean War, and named it after the Galata Bridge, which they crossed on their way to a coffeehouse to play cards.
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Biritch had many significant bridge-like developments: dealer chose the trump suit, or nominated his partner to do so; there was a call of "no trumps" ("biritch"); dealer's partner's hand became dummy; points were scored above and below the line; game was 3NT, 4 and 5 (although 8 club odd tricks and 15 spade odd tricks were needed); the score could be doubled and redoubled; and there were slam bonuses. It also has some features in common with solo whist. This game, and variants of it known as "bridge" and "bridge whist", became popular in the United States and the United Kingdom in the 1890s despite the long-established dominance of whist. Its breakthrough was its acceptance in 1894 by Lord Brougham at London's Portland Club.
In 1904, auction bridge was developed, in which the players bid in a competitive auction to decide the contract and declarer. The object became to make at least as many tricks as were contracted for, and penalties were introduced for failing to do so. In auction bridge, bidding beyond winning the auction is pointless; for example, if taking all 13 tricks, there is no difference in score between a 1 and a 7 final contract, as the bonus for rubber, small slam or grand slam depends on the number of tricks taken rather than the number of tricks contracted for.
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The modern game of contract bridge was the result of innovations to the scoring of auction bridge by Harold Stirling Vanderbilt and others. The most significant change was that only the tricks contracted for were scored below the line toward game or a slam bonus, a change that resulted in bidding becoming much more challenging and interesting. Also new was the concept of "vulnerability", which made sacrifices to protect the lead in a rubber more expensive. The various scores were adjusted to produce a more balanced and interesting game. Vanderbilt set out his rules in 1925, and within a few years contract bridge had so supplanted other forms of the game that "bridge" became synonymous with "contract bridge".
The form of bridge mostly played in clubs, tournaments and online is duplicate bridge. The number of people who play contract bridge has declined since its peak in the 1940s, when a survey found it was played in 44% of US households. The game is still widely played, especially amongst retirees, and in 2005 the ACBL estimated there were 25 million players in the US.
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Gameplay.
Overview.
Bridge is a four-player partnership trick-taking game with thirteen tricks per deal. The dominant variations of the game are rubber bridge, which is more common in social play; and duplicate bridge, which enables comparative scoring in tournament play. Each player is dealt thirteen cards from a standard 52-card deck. A starts when a player leads (i.e., plays the first card). The leader to the first trick is determined by the auction; the leader to each subsequent trick is the player who won the preceding trick. Each player, in clockwise order, plays one card on the trick. Players must play a card of the same suit as the original card led, unless they have none (said to be "void"), in which case they may play any card.
The rank of the cards played determines which player wins the trick. Within each suit, the ace is ranked highest followed by the king, queen and jack and then the ten through to the two. In a deal in which the auction has determined that there is no trump suit, the trick is won by the highest-ranked card of the suit led; cards of suits other than that led cannot win. In a deal with a trump suit, cards of that suit are superior in rank to any of the cards of any other suit. If one or more players plays a trump to a trick when void in the suit led, the highest-ranked trump wins. For example, if the trump suit is spades and a player is void in the suit led and plays a spade card, they win the trick if no other player plays a higher spade. If a card of the trump suit is led, the usual rule for trick-taking applies and the highest-ranked card of that suit wins.
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Unlike that of its predecessor, whist, the goal of bridge is not simply to take the most tricks in a deal. Instead, the goal is successfully to estimate how many tricks one's partnership can take, and then to meet or exceed that estimate. To illustrate this, the simpler partnership trick-taking game of spades has a similar mechanism: the usual trick-taking rules apply with the trump suit being spades, but in the beginning of the game, players "bid" or estimate how many tricks they can win, and the number of tricks bid by both players in a partnership are added. If a partnership takes at least that many tricks, they receive points for the round; otherwise, they lose penalty points.
Bridge extends the concept of bidding into an , in which partnerships compete to win a , specifying both how many tricks they will need to take in order to receive points and the trump suit (or "no trump", meaning that there will be no trump suit). Players take turns to call in a clockwise order: each player in turn either passes, doubleswhich increases the penalties for not making the contract specified by the opposing partnership's last bid, but also increases the reward for making itor redoubles, or states a contract that their partnership will adopt, which must be higher than the previous highest bid (if any). Eventually, the player who bid the highest contractwhich is determined by the contract's level as well as the trump suit or no trumpwins the contract for their partnership.
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In the example auction below, the east–west pair secures the contract of 6; the auction concludes when there have been three successive passes. Note that six tricks are added to stated contract values, so the six-level contract is a contract of twelve tricks. In practice, estimating a good contract without information about one's partner's hand is difficult, so there exist many bidding systems assigning meanings to bids, with common ones including Standard American, Acol, and 2/1 game forcing. Contrast with Spades, where players only have to bid their own hand.
After the contract is decided and the first lead is made, the declarer's partner (dummy) lays their cards face up on the table, and the declarer plays the dummy's cards as well as their own. The opposing partnership is called the , and their goal is to stop the declarer from fulfilling his contract. Once all the cards have been played, the deal is scored: if the declaring side makes their contract, they receive points based on the level of the contract, with some trump suits being worth more points than others and "no trump" being worth even more, as well as bonus points for any . If the declarer fails to fulfill the contract, the defenders receive points depending on the declaring side's undertricks (the number of tricks short of the contract) and whether the contract was "doubled" or "redoubled".
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Setup and dealing.
The four players sit in two partnerships with players sitting opposite their partners. A cardinal direction is assigned to each seat, so that one partnership sits in North and South, while the other sits in West and East. The cards may be freshly dealt or, in duplicate bridge games, pre-dealt. All that is needed in basic games are the cards and a method of keeping score, but there is often other equipment on the table, such as a board containing the cards to be played (in duplicate bridge), bidding boxes, or screens.
In rubber bridge each player draws a card at the start of the game; the player who draws the highest card deals first. The second highest card becomes the dealer's partner and takes the chair on the opposite side of the table. They play against the other two. The deck is shuffled and cut, usually by the player to the left of the dealer, before dealing. Players take turns to deal, in clockwise order. The dealer deals the cards clockwise, one card at a time. Normally, rubber bridge is played with two packs of cards and whilst one pack is being dealt, the dealer's partner shuffles the other pack. After shuffling the pack is placed on the right ready for the next dealer. Before dealing, the next dealer passes the cards to the previous dealer who cuts them.
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In duplicate bridge the cards are pre-dealt, either by hand or by a computerized dealing machine, in order to allow for competitive scoring. Once dealt, the cards are placed in a device called a "board", having slots designated for each player's cardinal direction seating position. After a deal has been played, players return their cards to the appropriate slot in the board, ready to be played by the next table.
Auction.
The dealer opens the auction and can make the first call, and the auction proceeds clockwise. When it is their turn to call, a player may passbut can enter into the bidding lateror bid a contract, specifying the level of their contract and either the trump suit or "no trump" (the denomination), provided that it is higher than the last bid by any player, including their partner. All bids promise to take a number of tricks in excess of six, so a bid must be between one (seven tricks) and seven (thirteen tricks). A bid is higher than another bid if either the level is greater (e.g., 2 over 1NT) or the denomination is higher at the same level, with the order being in ascending (or alphabetical) order: , , , , and NT (no trump) (e.g., 3 over 3). Calls may be made orally or with a bidding box.
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If the last bid was by the opposing partnership, one may also the opponents' bid, increasing the penalties for undertricks, but also increasing the reward for making the contract. Doubling does not carry to future bids by the opponents unless future bids are doubled again. A player on the opposing partnership being doubled may also , which increases the penalties and rewards further. Players may not see their partner's hand during the auction, only their own. There exist many bidding conventions that assign agreed meanings to various calls to assist players in reaching an optimal contract (or obstruct the opponents).
The auction ends when, after a player bids, doubles, or redoubles, every other player has passed, in which case the action proceeds to the play; or every player has passed and no bid has been made, in which case the round is considered to be "passed out" and not played.
Play.
The player from the declaring side who first bid the denomination named in the final contract becomes declarer. The player left to the declarer leads to the first trick. Dummy then lays his or her cards face-up on the table, organized in columns by suit. Play proceeds clockwise, with each player required to follow suit if possible. Tricks are won by the highest trump, or if there were none played, the highest card of the led suit. The player who won the previous trick leads to the next trick. The declarer has control of the dummy's cards and tells his partner which card to play at dummy's turn. There also exist conventions that communicate further information between defenders about their hands during the play.
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At any time, a player may , stating that their side will win a specific number of the remaining tricks. The claiming player lays his cards down on the table and explains the order in which he intends to play the remaining cards. The opponents can either accept the claim and the round is scored accordingly, or dispute the claim. If the claim is disputed, play continues with the claiming player's cards face up in rubber games, or in duplicate games, play ceases and the tournament director is called to adjudicate the hand.
Scoring.
At the end of the hand, points are awarded to the declaring side if they make the contract, or else to the defenders. Partnerships can be , increasing the rewards for making the contract, but also increasing the penalties for undertricks. In rubber bridge, if a side has won 100 contract points, they have won a and are vulnerable for the remaining rounds, but in duplicate bridge, vulnerability is predetermined based on the number of each board.
If the declaring side makes their contract, they receive points for , or tricks bid and made in excess of six. In both rubber and duplicate bridge, the declaring side is awarded 20 points per odd trick for a contract in clubs or diamonds, and 30 points per odd trick for a contract in hearts or spades. For a contract in notrump, the declaring side is awarded 40 points for the first odd trick and 30 points for the remaining odd tricks. Contract points are doubled or quadrupled if the contract is respectively doubled or redoubled.
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In rubber bridge, a partnership wins one game once it has accumulated 100 contract points; excess contract points do not carry over to the next game. A partnership that wins two games wins the rubber, receiving a bonus of 500 points if the opponents have won a game, and 700 points if they have not.
Overtricks score the same number of points per odd trick, although their doubled and redoubled values differ. Bonuses vary between the two bridge variations both in score and in type (for example, rubber bridge awards a bonus for holding a certain combination of high cards), although some are common between the two.
A larger bonus is awarded if the declaring side makes a small slam or grand slam, a contract of 12 or 13 tricks respectively. If the declaring side is not vulnerable, a small slam gets 500 points, and a grand slam 1000 points. If the declaring side is vulnerable, a small slam is 750 points and a grand slam is 1,500.
In rubber bridge, the rubber finishes when a partnership has won two games, but the partnership receiving the most "overall" points wins the rubber. Duplicate bridge is scored comparatively, meaning that the score for the hand is compared to other tables playing the same cards and match points are scored according to the comparative results: usually either "matchpoint scoring", where each partnership receives 2 points (or 1 point) for each pair that they beat, and 1 point (or point) for each tie; or IMPs (international matchpoint) scoring, where the number of IMPs varies (but less than proportionately) with the points difference between the teams.
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Undertricks are scored in both variations as follows:
Rules.
The rules of the game are referred to as the "laws" as promulgated by various bridge organizations.
Duplicate.
The official rules of duplicate bridge are promulgated by the WBF as "The Laws of Duplicate Bridge 2017". The Laws Committee of the WBF, composed of world experts, updates the Laws every 10 years; it also issues a Laws Commentary advising on interpretations it has rendered.
In addition to the basic rules of play, there are many additional rules covering playing conditions and the rectification of irregularities, which are primarily for use by tournament directors who act as referees and have overall control of procedures during competitions. But various details of procedure are left to the discretion of the zonal bridge organisation for tournaments under their aegis and some (for example, the choice of "movement") to the sponsoring organisation (for example, the club).
Some zonal organisations of the WBF also publish editions of the Laws. For example, the American Contract Bridge League (ACBL) publishes the "Laws of Duplicate Bridge" and additional documentation for club and tournament directors.
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Rubber.
There are no universally accepted rules for rubber bridge, but some zonal organisations have published their own. An example for those wishing to abide by a published standard is "The Laws of Rubber Bridge" as published by the American Contract Bridge League.
The majority of rules mirror those of duplicate bridge in the bidding and play and differ primarily in procedures for dealing and scoring.
Online.
In 2001, the WBF promulgated a set of laws for online play.
Tournaments.
Bridge is a game of skill played with randomly dealt cards, which makes it also a game of chance, or more exactly, a tactical game with inbuilt randomness, imperfect knowledge and restricted communication. The chance element is in the deal of the cards; in duplicate bridge some of the chance element is eliminated by comparing results of multiple pairs in identical situations. This is achievable when there are eight or more players, sitting at two or more tables, and the deals from each table are preserved and passed to the next table, thereby "duplicating" them for the other table(s) of players. At the end of a session, the scores for each deal are compared, and the most points are awarded to the players doing the best with each particular deal. This measures relative skill (but still with an element of luck) because each pair or team is being judged only on the ability to bid with, and play, the same cards as other players.
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Duplicate bridge is played in clubs and tournaments, which can gather as many as several hundred players. Duplicate bridge is a mind sport, and its popularity gradually became comparable to that of chess, with which it is often compared for its complexity and the mental skills required for high-level competition. Bridge and chess are the only "mind sports" recognized by the International Olympic Committee, although they were not found eligible for the main Olympic program. In October 2017 the British High Court ruled against the English Bridge Union, finding that Bridge is not a sport under a definition of sport as involving physical activity, but did not rule on the "broad, somewhat philosophical question" as to whether or not bridge is a sport.
The basic premise of duplicate bridge had previously been used for whist matches as early as 1857. Initially, bridge was not thought to be suitable for duplicate competition; it was not until the 1920s that (auction) bridge tournaments became popular.
In 1925 when contract bridge first evolved, bridge tournaments were becoming popular, but the rules were somewhat in flux, and several different organizing bodies were involved in tournament sponsorship: the American Bridge League (formerly the "American Auction Bridge League", which changed its name in 1929), the American Whist League, and the United States Bridge Association. In 1937, the first officially recognized world championship was held in Budapest. In 1958, the World Bridge Federation (WBF) was founded to promote bridge worldwide, coordinate periodic revision to the Laws (each ten years, next in 2027) and conduct world championships.
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Bidding boxes and screens.
In tournaments, "bidding boxes" are frequently used, as noted above. These avoid the possibility of players at other tables hearing any spoken bids. The bidding cards are laid out in sequence as the auction progresses. Although it is not a formal rule, many clubs adopt a protocol that the bidding cards stay revealed until the first playing card is tabled, after which point the bidding cards are put away. Bidding pads are an alternative to bidding boxes. A bidding pad is a block of 100mm square tear-off sheets. Players write their bids on the top sheet. When the first trick is complete the sheet is torn off and discarded.
In top national and international events, "bidding screens" are used. These are placed diagonally across the table, preventing partners from seeing each other during the game; often the screen is removed after the auction is complete.
Strategy.
Bidding.
Much of the complexity in bridge arises from the difficulty of arriving at a good final contract in the auction (or deciding to let the opponents declare the contract). This is a difficult problem: the two players in a partnership must try to communicate enough information about their hands to arrive at a makeable contract, but the information they can exchange is restricted – information may be passed only by the calls made and later by the cards played, not by other means; in addition, the agreed-upon meaning of each call and play must be available to the opponents.
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Since a partnership that has freedom to bid gradually at leisure can exchange more information, and since a partnership that can interfere with the opponents' bidding (as by raising the bidding level rapidly) can cause difficulties for their opponents, bidding systems are both informational and strategic. It is this mixture of information exchange and evaluation, deduction, and tactics that is at the heart of bidding in bridge.
A number of basic rules of thumb in bridge bidding and play are summarized as bridge maxims.
Systems and conventions.
A "bidding system" is a set of partnership agreements on the meanings of bids. A partnership's bidding system is usually made up of a core system, modified and complemented by specific conventions (optional customizations incorporated into the main system for handling specific bidding situations) which are pre-chosen between the partners prior to play. The line between a well-known convention and a part of a system is not always clear-cut: some bidding systems include specified conventions by default. Bidding systems can be divided into mainly natural systems such as Acol and Standard American, and mainly artificial systems such as the Precision Club and Polish Club.
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Calls are usually considered to be either "natural" or "conventional" (artificial). A natural call carries a meaning that reflects the call; a natural bid intuitively showing hand or suit strength based on the level or suit of the bid, and a natural double expressing that the player believes that the opposing partnership will not make their contract. By contrast, a conventional (artificial) call offers and/or asks for information by means of pre-agreed coded interpretations, in which some calls convey very specific information or requests that are not part of the natural meaning of the call. Thus in response to 4NT, a 'natural' bid of 5 would state a preference towards a diamond suit or a desire to play in five diamonds, whereas if the partners have agreed to use the common Blackwood convention, a bid of 5 in the same situation would say nothing about the diamond suit, but would tell the partner that the hand in question contains exactly one ace.
Conventions are valuable in bridge because of the need to pass information beyond a simple like or dislike of a particular suit, and because the limited bidding space can be used more efficiently by adopting a conventional (artificial) meaning for a given call where a natural meaning has less utility, because the information it conveys is not valuable or because the desire to convey that information arises only rarely. The conventional meaning conveys more useful (or more frequently useful) information. There are a very large number of conventions from which players can choose; many books have been written detailing bidding conventions. Well-known conventions include Stayman (to ask the opening 1NT bidder to show any four-card major suit), Jacoby transfers (a request by (usually) the weak hand for the partner to bid a particular suit first, and therefore to become the declarer), and the Blackwood convention (to ask for information on the number of aces and kings held, used in slam bidding situations).
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The term "preempt" refers to a high-level tactical bid by a weak hand, relying upon a very long suit rather than high cards for tricks. Preemptive bids serve a double purpose – they allow players to indicate they are bidding on the basis of a long suit in an otherwise weak hand, which is important information to share, and they also consume substantial bidding space which prevents a possibly strong opposing pair from exchanging information on their cards. Several systems include the use of opening bids or other early bids with weak hands including long (usually six to eight card) suits at the 2, 3 or even 4 or 5 levels as preempts.
Basic natural systems.
As a rule, a natural suit bid indicates a holding of at least four (or more, depending on the situation and the system) cards in that suit as an opening bid, or a lesser number when supporting partner; a natural NT bid indicates a balanced hand.
Most systems use a count of high card points as the basic evaluation of the strength of a hand, refining this by reference to shape and distribution if appropriate. In the most commonly used point count system, aces are counted as 4 points, kings as 3, queens as 2, and jacks as 1 point; therefore, the deck contains 40 points. In addition, the "distribution" of the cards in a hand into suits may also contribute to the strength of a hand and be counted as distribution points. A better than average hand, containing 12 or 13 points, is usually considered sufficient to "open" the bidding, i.e., to make the first bid in the auction. A combination of two such hands (i.e., 25 or 26 points shared between partners) is often sufficient for a partnership to bid, and generally to make, game in a major suit or notrump (more are usually needed for a minor suit game, as the level is higher).
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