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Peloponnese

The Peloponnese, Peloponnesus (Πελοπόννησος, ) or Morea (Μωριάς) is a peninsula and geographic region in Southern Greece, and the southernmost region of the Balkans. It is connected to the central part of the country by the Isthmus of Corinth land bridge which separates the Gulf of Corinth from the Saronic Gulf. From the late Middle Ages until the 19th century, the peninsula was known as the Morea, a name still in colloquial use in its demotic form.

The peninsula is divided among three administrative regions: most belongs to the Peloponnese region, with smaller parts belonging to the West Greece and Attica regions.

Geography
The Peloponnese is a peninsula located at the southern tip of the mainland, 21,549.6 sqkm in area, and constitutes the southernmost part of mainland Greece. It is connected to the mainland by the Isthmus of Corinth, where the Corinth Canal was constructed in 1893. However, it is also connected to the mainland by several bridges across the canal, including two submersible bridges at the north and the south end. Near the northern tip of the peninsula, there is another bridge, the Rio–Antirrio bridge (completed 2004). The peninsula has a mountainous interior and deeply indented coasts. The Peloponnese possesses four south-pointing peninsulas, the Messenian, the Mani, the Cape Malea (also known as Epidaurus Limera), and the Argolid in the far northeast of the Peloponnese. Mount Taygetus in the south is the highest mountain in the Peloponnese, at 2407 m. Οther important mountains include Cyllene in the northeast (2376 m), Aroania in the north (2355 m), Erymanthos (2224 m) and Panachaikon in the northwest (1926 m), Mainalon in the center (1981 m), and Parnon in the southeast (1935 m). The entire peninsula is earthquake prone and has been the site of many earthquakes in the past.

The longest river is the Alfeios in the west (110 km), followed by the Evrotas in the south (82 km), and also the Pineios, also in the west (70 km). Extensive lowlands are found only in the west, except for the Evrotas Valley in the south and the Argolid in the northeast. The Peloponnese is home to numerous spectacular beaches, which are a major tourist draw.

Two groups of islands lie off the Peloponnesian coast: the Argo-Saronic Islands to the east, and the Ionian to the west. The island of Kythira, off the Epidaurus Limeira peninsula to the south of the Peloponnese, is considered to be part of the Ionian Islands. The island of Elafonisos used to be part of the peninsula but was separated following the major quake of 365 AD.

Since antiquity, and continuing to the present day, the Peloponnese has been divided into seven major regions: Achaea (north), Corinthia (northeast), Argolis (east), Arcadia (center), Laconia (southeast), Messenia (southwest), and Elis (west). Each of these regions is headed by a city. The largest city is Patras (pop. 170,000) in Achaia, followed by Kalamata (pop. 55,000) in Messenia.

Climate
Peloponnese for the most part enjoys a hot-summer Mediterranean climate (Köppen climate classification: Csa), while the Gulf of Corinth has a hot semi-arid climate (Köppen climate classification: BSh). Rainfall is higher in the west coasts, while the east of the peninsula is significantly drier. Average annual temperatures can reach up to 20.2 C while summer highs reach up to 36.0 C in Sparta, within the Evrotas Valley. On 27 June 2007, Monemvasia registered a staggering minimum temperature of 35.9°C, which is the highest minimum temperature ever recorded in mainland Greece and Continental Europe.

Mythology and early history
The peninsula has been inhabited since prehistoric times. Its modern name derives from ancient Greek mythology, specifically the legend of the hero Pelops, who was said to have conquered the entire region. The name Peloponnesos means Island or Peninsula (archaic meaning of the word ) of Pelops.

The Mycenaean civilization, mainland Greece's (and Europe's) first major civilization, dominated the Peloponnese in the Bronze Age from the palaces of Mycenae, Pylos and Tiryns; among others. The Mycenaean civilization collapsed suddenly at the end of the 2nd millennium BC. Archeological research has found that many of its cities and palaces show signs of destruction. The subsequent period, known as the Greek Dark Ages, is marked by an absence of written records.

Classical antiquity
In 776 BC, the first Olympic Games were held at Olympia, in the western Peloponnese and this date is sometimes used to denote the beginning of the classical period of Greek antiquity. During classical antiquity, the Peloponnese was at the heart of the affairs of ancient Greece, possessed some of its most powerful city-states, and was the location of some of its bloodiest battles.

The major cities of Sparta, Corinth, Argos and Megalopolis were all located on the Peloponnese, and it was the homeland of the Peloponnesian League. Soldiers from the peninsula fought in the Persian Wars, and it was also the scene of the Peloponnesian War of 431–404 BC. The entire Peloponnese with the notable exception of Sparta joined Alexander's expedition against the Persian Empire.

Along with the rest of Greece, the Peloponnese fell to the expanding Roman Republic in 146 BC, when the Romans razed the city of Corinth and massacred its inhabitants. The Romans created the province of Achaea comprising the Peloponnese and central Greece. During the Roman period, the peninsula remained prosperous but became a provincial backwater, relatively cut off from the affairs of the wider Roman world.

Byzantine rule
After the partition of the Empire in 395, the Peloponnese became a part of the East Roman or Byzantine Empire. The devastation of Alaric's raid in 396–397 led to the construction of the Hexamilion wall across the Isthmus of Corinth. Through most of late antiquity, the peninsula retained its urbanized character: in the 6th century, Hierocles counted 26 cities in his Synecdemus. By the latter part of that century, however, building activity seems to have stopped virtually everywhere except Constantinople, Thessalonica, Corinth, and Athens. This has traditionally been attributed to calamities such as plague, earthquakes and Slavic invasions. However, more recent analysis suggests that urban decline was closely linked with the collapse of long-distance and regional commercial networks that underpinned and supported late antique urbanism in Greece, as well as with the generalized withdrawal of imperial troops and administration from the Balkans.

Slavic invasion, settlement and decline
The scale of the Slavic invasion and settlement in the 7th and 8th centuries remains a matter of dispute, although it is nowadays considered much smaller than previously thought. The Slavs did occupy most of the peninsula, as evidenced by the abundance of Slavic toponyms, but these toponyms accumulated over centuries rather than as a result of an initial "flood" of Slavic invasions, and many appear to have been mediated by speakers of Greek, or in mixed Slavic-Greek compounds.

Fewer Slavic toponyms appear on the eastern coast, which remained in Byzantine hands and was included in the thema of Hellas, established by Justinian II c. 690. While traditional historiography has dated the arrival of Slavs to southern Greece to the late 6th century, according to Florin Curta there is no evidence for a Slavic presence in the Peloponnese until after c. 700 AD, when Slavs may have been allowed to settle in specific areas that had been depopulated.

Relations between the Slavs and Greeks were probably peaceful apart from intermittent uprisings. There was also a continuity of the Peloponnesian Greek population. This is especially true in Mani and Tsakonia, where Slavic incursions were minimal, or non-existent. Being agriculturalists, the Slavs probably traded with the Greeks, who remained in the towns, while Greek villages continued to exist in the interior, governing themselves, possibly paying tribute to the Slavs. The first attempt by the Byzantine imperial government to re-assert its control over the independent Slavic tribes of the Peloponnese occurred in 783, with the logothete Staurakios' overland campaign from Constantinople into Greece and the Peloponnese, which according to Theophanes the Confessor made many prisoners and forced the Slavs to pay tribute.

From the mid-9th century, following a Slavic revolt and attack on Patras, a determined Hellenization process was carried out. According to the Chronicle of Monemvasia, in 805 the Byzantine governor of Corinth went to war with the Slavs, exterminated them, and allowed the original inhabitants to claim their lands. They regained control of the city of Patras and the region was re-settled with Greeks. Many Slavs were transported to Asia Minor, and many Asian, Sicilian and Calabrian Greeks were resettled in the Peloponnese. By the turn of the 9th century, the entire Peloponnese was formed into the new thema of Peloponnesos, with its capital at Corinth.

The imposition of Byzantine rule over the Slavic enclaves may have largely been a process of Christianization and accommodation of Slavic chieftains into the Imperial fold, as literary, epigraphic and sigillographic evidence testify to Slavic archontes participating in Imperial affairs. By the end of the 9th century, the Peloponnese was culturally and administratively Greek again, except for a few small Slavic tribes in the mountains such as the Melingoi and Ezeritai. Although they were to remain relatively autonomous until Ottoman times, such tribes were the exception rather than the rule. Even the Melingoi and Ezeritai, however, could speak Greek and appear to have been Christian.

The success of the Hellenization campaign also shows that the Slavs had settled among many Greeks, in contrast to areas further north in what is now Bulgaria and the former Yugoslavia, as those areas could not be Hellenized when they were recovered by the Byzantines in the early 11th century. A human genetics study in 2017 showed that the Peloponnesians have little admixture with populations of the Slavic homeland and are much closer to Sicilians and southern Italians.

Apart from the troubled relations with the Slavs, the coastal regions of the Peloponnese suffered greatly from repeated Arab raids following the Arab capture of Crete in the 820s and the establishment of a corsair emirate there. After the island was recovered by Byzantium in 961 however, the region entered a period of renewed prosperity, where agriculture, commerce, and urban industry flourished.

Frankish rule and Byzantine reconquest

In 1205, following the destruction of the Byzantine Empire by the forces of the Fourth Crusade, the Crusaders under William of Champlitte and Geoffrey of Villehardouin marched south through mainland Greece and conquered the Peloponnese against sporadic local Greek resistance. The Franks then founded the Principality of Achaea, nominally a vassal of the Latin Empire, while the Venetians occupied several strategically important ports around the coast such as Navarino and Coron, which they retained into the 15th century. The Franks popularized the name Morea for the peninsula, which first appears as the name of a small bishopric in Elis during the 10th century. Its etymology is disputed, but it is most commonly held to be derived from the mulberry tree (morea), whose leaves are similar in shape to the peninsula.

In 1208, William I founded a commission at Andravida consisted of Latin bishops, two bannerets and five Greek magnates and chaired by himself, to assess the land and divide it, according to Latin practice, in fiefs. The result was divide the country into twelve baronies, mostly centred around a newly constructed castle—a testament to the fact that the Franks were a military elite amidst a potentially hostile Greek population. The twelve temporal barons were joined by seven ecclesiastic lords, headed by the Latin Archbishop of Patras. Each of the latter was granted a number of estates as knightly fiefs, with the Archbishop receiving eight, the other bishops four each, and likewise four granted to each of the military orders: the Templars, Hospitallers and the Teutonic Knights. Shortly after 1260, a thirteenth barony, that of Arcadia (modern Kyparissia) was established, which was also a personal fief of the Villehardouins. The barons retained considerable powers and privileges, so that the Prince was not an absolute sovereign but rather a "first among equals" among them. Thus they had the right to construct a castle without the Prince's permission, or to decree capital punishment. Since Salic Law was not adopted in Achaea, women could also inherit the fiefs.

Despotate of Morea and Ottoman incursions
Frankish supremacy in the peninsula, however, received a critical blow after the Battle of Pelagonia, when William II of Villehardouin was forced to cede the newly constructed fortress and palace at Mystras near ancient Sparta to a resurgent Byzantium. At this point, the emperor concluded an agreement with the captive prince: William and his men would be set free in exchange for an oath of fealty, and for the cession of Monemvasia, Grand Magne, and Mystras. The handover was effected in 1262, and henceforth Mystras was the seat of the governor of the Byzantine territories in the Morea. Initially this governor (kephale) was changed every year, but after 1308 they started being appointed for longer terms. Almost immediately on his return to the Morea, William of Villehardouin renounced his oath to the emperor, and warfare broke out between Byzantines and Franks. The first Byzantine attempts to subdue the Principality of Achaea were beaten back in the battles of Prinitsa and Makryplagi, but the Byzantines were firmly ensconced in Laconia. Warfare became endemic, and the Byzantines slowly pushed the Franks back. The insecurity engendered by the raids and counter-raids caused the inhabitants of Lacedaemon to abandon their exposed city and settle at Mystras, in a new town built under the shadow of the fortress.

While Mystras served as the provincial capital from this time, it became a royal capital in 1349, when the first despot was appointed to rule over the Morea. The Byzantine Emperor John VI Kantakouzenos, reorganized the territory in 1349 to establish it as an appanage for his son, the Despot Manuel Kantakouzenos. Around that time, the Ottoman Turks began raiding the Peloponnese, but their raids intensified only after 1387 when the energetic Evrenos Bey took control. Exploiting the quarrels between Byzantines and Franks, he plundered across the peninsula and forced both the Byzantine despots and the remaining Frankish rulers to acknowledge Ottoman suzerainty and pay tribute. This situation lasted until the Ottoman defeat at the Battle of Ankara in 1402, after which Ottoman power was for a time checked. From 1349 until its surrender to the Ottoman Turks on 31 May 1460, Mystras was the residence of a Despot who ruled over the Byzantine Morea, known as the "Despotate of the Morea". For the larger portion of his reign, Manuel maintained peaceful relations with his Latin neighbors and secured a long period of prosperity for the area. Greco-Latin cooperation included an alliance to contain the raids of the Ottoman Sultan Murad I into Morea in the 1360s. The rival Palaiologos dynasty seized the Morea after Manuel's death in 1380, with Theodore I Palaiologos becoming despot in 1383.

Theodore I ruled until 1407, consolidating Byzantine rule and coming to terms with his more powerful neighbours—particularly the expansionist Ottoman Empire, whose suzerainty he recognised. Subsequent despots were the sons of the Emperor Manuel II Palaiologos, brother of the despot Theodore: Theodore II, Constantine, Demetrios, and Thomas. As Latin power in the Peloponnese waned during the 15th century, the Despotate of the Morea expanded to incorporate the entire peninsula in 1430 with territory being acquired by dowry settlements, and the conquest of Patras by Constantine. However, in 1446 the Ottoman Sultan Murad II destroyed the Byzantine defences—the Hexamilion wall at the Isthmus of Corinth. His attack opened the peninsula to invasion, though Murad died before he could exploit this. His successor Mehmed II "the Conqueror" captured the Byzantine capital Constantinople in 1453. The despots, Demetrios Palaiologos and Thomas Palaiologos, brothers of the last emperor, failed to send him any aid, as Morea was recovering from a recent Ottoman attack. Their own incompetence resulted in the Morea revolt of 1453–1454 led by Manuel Kantakouzenos against them, during which they invited in Ottoman troops to help them put down the revolt. At this time, the Greek archons made peace with Mehmed. After more years of incompetent rule by the despots, their failure to pay their annual tribute to the Sultan, and finally their own revolt against Ottoman rule, Mehmed came into the Morea in May 1460. Demetrios ended up a prisoner of the Ottomans and his younger brother Thomas fled. By the end of the summer the Ottomans had achieved the submission of virtually all cities possessed by the Greeks.

Ottoman incursions into the Morea resumed under Turahan Bey after 1423. Despite the reconstruction of the Hexamilion wall at the Isthmus of Corinth, the Ottomans under Murad II breached it in 1446, forcing the Despots of the Morea to re-acknowledge Ottoman suzerainty, and again under Turahan in 1452 and 1456. Following the occupation of the Duchy of Athens in 1456, the Ottomans occupied a third of the Peloponnese in 1458, and Sultan Mehmed II extinguished the remnants of the Despotate in 1460. A few holdouts remained for a time. The rocky peninsula of Monemvasia refused to surrender, and it was first ruled for a brief time by a Catalan corsair. When the population drove him out, they obtained the consent of Thomas to submit to the Pope's protection before the end of 1460. The Mani Peninsula at the south end of the Morea resisted under a loose coalition of the local clans, and that area then came under Venice's rule. The last holdout was Salmeniko, in the Morea's northwest. Graitzas Palaiologos was the military commander there, stationed at Salmeniko Castle. While the town eventually surrendered, Graitzas and his garrison and some town residents held out in the castle until July 1461, when they escaped and reached Venetian territory. Only the Venetian fortresses of Modon, Coron, Navarino, Monemvasia, Argos and Nauplion escaped Ottoman control.

Albanian migration, settlement and relocations to Italy
The same period was also marked by the migration and settlement of Christian Albanians to parts of Central Greece and the Peloponnese, a group that eventually became known as the Arvanites The Albanians settled in successive waves, often invited by the local rulers. They start appearing more frequently in the historical record from during the second part of the 14th century, when they were being offered arable land, pasture and favorable taxation in exchange for military service. One of the larger groups of Albanian settlers, amounting to 10,000, settled the Peloponnese during the reign of Theodore I Palaiologos, first in Arcadia and subsequently in other regions around Messenia, Argolis, Elis and Achaia. Around 1418, a second large group arrived, possibly fleeing Aetolia, Acarnania and Arta, where Albanian political power had been defeated. The settling Albanians lived in tribes spread out into small villages, practicing nomadic lifestyles based on pastoralism and animal husbandry. By the mid-15th century, they formed a substantial part of the population of the Peloponnesus. Military sources of the era (1425) report about 30,000 Albanian men who could carry arms in the Peloponnese. The Greeks tended to live in large villages and cities, while Albanians in small villages.

Following Ottoman conquest, many Albanians fled to Italy, settling primarily in nowadays Arbereshe villages of Calabria and Sicily. On the other hand, in an effort to control the remaining Albanians, during the second half of the 15th century, the Ottomans adopted favorable tax policies towards them, likely in continuation of similar Byzantine practices. This policy had been discontinued by the early 16th century. Throughout the Ottoman–Venetian wars, many Albanians died or were captured in service to the Venetians; at Nafpaktos, Nafplio, Argos, Methoni, Koroni and Pylos. Furthermore, 8,000 Albanian stratioti, most of them along with their families, left the Peloponnese to continue their military service under the Republic of Venice or the Kingdom of Naples. At the end of the Ottoman–Venetian wars, a large number of Albanians had fled from the Peloponnese to Sicily.

A demographic census by Alfred Philippson, based on fieldwork between 1887 and 1889, found that out of the approximately 730,000 inhabitants of the Peloponnese, and the three neighboring islands of Poros, Hydra and Spetses, Arvanites numbered 90,253, or 12.3% of the total population.

Ottoman conquest, Venetian interlude and Ottoman reconquest
The Venetian fortresses were conquered in a series of Ottoman-Venetian Wars: the first war, lasting from 1463 to 1479, saw much fighting in the Peloponnese, resulting in the loss of Argos, while Modon and Coron fell in 1500 during the second war. Coron and Patras were captured in a crusading expedition in 1532, led by the Genoese admiral Andrea Doria, but this provoked another war in which the last Venetian possessions on the Greek mainland were lost.

Following the Ottoman conquest, the peninsula was made into a province (sanjak), with 109 ziamets and 342 timars. During the first period of Ottoman rule (1460–1687), the capital was first in Corinth (Turk. Gördes), later in Leontari (Londari), Mystras (Misistire) and finally in Nauplion (Tr. Anaboli). Sometime in the mid-17th century, the Morea became the centre of a separate eyalet, with Patras (Ballibadra) as its capital. Until the death of Suleiman the Magnificent in 1570, the Christian population (counted at some 42,000 families c. 1550 ) managed to retain some privileges and Islamization was slow, mostly among the Albanians or the estate owners who were integrated into the Ottoman feudal system.

Although they quickly came to control most of the fertile lands, Muslims remained a distinct minority. Christian communities retained a large measure of self-government, but the entire Ottoman period was marked by a flight of the Christian population from the plains to the mountains. This occasioned the rise of the klephts, armed brigands and rebels, in the mountains, as well as the corresponding institution of the government-funded armatoloi to check the klephts' activities.

With the outbreak of the "Great Turkish War" in 1683, the Venetians under Francesco Morosini occupied the entire peninsula by 1687, and received recognition by the Ottomans in the Treaty of Karlowitz (1699). The Venetians established their province as the "Kingdom of the Morea" (It. Regno di Morea), but their rule proved unpopular, and when the Ottomans invaded the peninsula in 1715, most local Greeks welcomed them. The Ottoman reconquest was easy and swift, and was recognized by Venice in the Treaty of Passarowitz in 1718.

The Peloponnese now became the core of the Morea Eyalet, headed by the Mora valesi, who until 1780 was a pasha of the first rank (with three horsetails) and held the title of vizier. After 1780 and until the Greek War of Independence, the province was headed by a muhassil. The pasha of the Morea was aided by several subordinate officials, including a Christian translator (dragoman), who was the senior Christian official of the province. As during the first Ottoman period, the Morea was divided into 22 districts or beyliks. The capital was first at Nauplion, but after 1786 at Tripolitza (Tr. Trabliçe).

The Greeks of the Peloponnese rose against the Ottomans with Russian aid during the so-called "Orlov Revolt" of 1770, but it was swiftly and brutally suppressed by bands of Muslim Albanian mercenaries hired by the Ottomans. Referred to by the local Greek populace as "Turk-Albanians", those forces had also destroyed many cities and towns in Epirus during the 1769–70 revolt there. The Peloponnese suffered more than any other Greek inhabited area by irregular Albanian gangs during the decades following. In Patras nearly no one was left alive after the Turkish-Albanian invasion. The city of Mystras was left in ruins and the metropolitan bishop Ananias was executed despite having saved the life of several Turks during the uprising. A great number of local Greeks were killed by the Albanian groups, while children were sold to slavery. It is estimated that 20,000 local Greeks were captured during those nine years of devastation by those Albanian mercenaries and sold to slave markets. Also an additional of 50,000 Greeks left Peloponesse: around one-sixth of the pre-1770 population.

The Ottoman government was unable to pay the wages the Albanian mercenaries demanded for their service, causing the latter to ravage the region even after revolt had been put down. 1770-1779 was a prolonged period of devastation and atrocities committed by Albanian irregulars in the Peloponnese. In 1774 the Russo-Turkish War ended with the Treaty of Küçük Kaynarca which granted general amnesty to the population. Nevertheless, attacks by Muslim Albanian mercenaries in the region continued not only against the Greek population but also against Turks. The extensive destruction and lack of control in the Peloponnese forced the central Ottoman government to send a regular Turkish military force to suppress those Albanian troops in 1779, and eventually drive them out from Peloponnese. As a result of the invasion by those mercenary groups the local population had to found refuge in the mountains of Peloponnese to avoid persecution. The total population decreased during this time, while the Muslim element in it increased.

As such Greek resistance in the peninsula was reinforced and powerful groups of klephts were formed under the clans of Zacharias, Melios, Petmezas and Kolokotronis. Klephtic songs of that era describe the resistance activities. Nevertheless, through the privileges granted with the Treaty of Kuchuk-Kainarji, especially the right for the Christians to trade under the Russian flag, led to a considerable economic flowering of the local Greeks, which, coupled with the increased cultural contacts with Western Europe (Modern Greek Enlightenment) and the inspiring ideals of the French Revolution, laid the groundwork for the Greek War of Independence.

Modern Greece

The Peloponnesians played a major role in the Greek War of Independence – the war began in the Peloponnese, when rebels took control of Kalamata on March 23, 1821. After the arrival of Ypsilantis's emissaries, local people rose under the leadership of Mavromichalis. Greek and Albanian insurgents organised in units of armed civilians took control of most of the fortresses. The Greek insurgents made rapid progress and the entire peninsula was under Greek control within a few months, except for a few coastal forts and the main Turkish garrison at Tripolitsa. The fighting was fierce and marked by atrocities on both sides; eventually the entire Muslim population was either massacred or fled to the forts. The capture of Tripolitsa in September 1821 marked a turning point. Rivalries among the insurgents eventually erupted into civil war in 1824, which enabled the Ottoman Egyptian vassal Ibrahim Pasha to land in the peninsula in 1825.

The Peloponnese peninsula was the scene of fierce fighting and extensive devastation following the arrival of Ibrahim's Egyptian troops. Partly as a result of the atrocities committed by Ibrahim, the UK, France, and the Russian Empire decided to intervene in favor of the Greeks. The decisive naval Battle of Navarino was fought in 1827 off Pylos on the west coast of the Peloponnese, where a combined British, French and Russian fleet decisively defeated the Turko-Egyptian fleet. Subsequently, a French expeditionary corps cleared the last Turko-Egyptian forces from the peninsula in 1828. The city of Nafplion, on the east coast of the peninsula, became the first capital of the independent Greek state. By the conclusion of the war, the entire Muslim population of the newly independent Greek state, including the Peloponnese, had been exterminated or had fled.

During the 19th and early 20th centuries, the region became relatively poor and economically isolated. A significant part of its population emigrated to the larger cities of Greece, especially Athens, and other countries such as the United States and Australia. It was badly affected by the Second World War and Greek Civil War, experiencing some of the worst atrocities committed in Greece during those conflicts. Living standards improved dramatically throughout Greece after the country accedes to the European Union in 1981.

The Corinth Canal was completed in the late 19th century, linking the Aegean Sea with the Gulf of Corinth and the Ionian. In 2001, the Rio-Antirio Bridge was completed, linking the western Peloponnese to western Greece. In late August 2007, large parts of Peloponnese suffered from wildfires, which caused severe damage in villages and forests and the death of 77 people. The impact of the fires to the environment and economy of the region are still unknown. It is thought to be one of the largest environmental disasters in modern Greek history.

Regional units

 * Arcadia – 100,611 inhabitants
 * Argolis – 108, 636 inhabitants
 * Corinthia – 144,527 inhabitants (except municipalities of Agioi Theodoroi and most of Loutraki-Perachora, which lie east of the Corinth Canal)
 * Laconia – 100,871 inhabitants
 * Messenia – 180,264 inhabitants
 * Achaea – 331,316 inhabitants
 * Elis – 198,763 inhabitants
 * Islands (only the municipality Troizinia and part of Poros)

Cities
The principal modern cities of the Peloponnese are (2011 census):
 * Patras – 170,896 inhabitants
 * Kalamata – 62,409 inhabitants
 * Corinth – 38,132 inhabitants
 * Tripoli – 30,912 inhabitants
 * Aigio – 26,523 inhabitants
 * Pyrgos – 25,180 inhabitants
 * Argos – 24,700 inhabitants
 * Sparta – 19,854 inhabitants
 * Nafplio – 18,910 inhabitants
 * Amaliada - 18,303 inhabitants

Archaeological sites

The Peloponnese possesses many important archaeological sites dating from the Bronze Age through to the Middle Ages. Among the most notable are:
 * Bassae‡ (ancient town and the temple of Epikourios Apollo and Greece's first UNESCO World Heritage Site)
 * Corinth (ancient city)
 * Diros caves (4000 – 3000 BC)
 * Epidaurus‡ (ancient religious and healing centre)
 * Koroni (medieval seaside fortress and city walls)
 * Kalamata Acropolis (medieval acropolis and fortress located within the modern city)
 * Messene (ancient city)
 * Methoni (medieval seaside fortress and city walls)
 * Mystras‡ (medieval Byzantine fortress-town near Sparta)
 * Monemvasia (medieval Byzantine fortress-town)
 * Mycenae‡ (fortress-town of the eponymous civilization)
 * Olympia‡ (site of the Ancient Olympic Games)
 * Pylos (the Palace of Nestor and a well-preserved medieval/early modern fortress)
 * Pavlopetri (the oldest underwater city in the world, located in Vatika Bay, dating from the early Bronze Age 3500 BC)
 * Sparta
 * Tegea (ancient religious centre)
 * Tiryns‡ (ancient fortified settlement)

‡ UNESCO World Heritage Site

Cuisine
Specialities of the region:
 * Gogges (or Gogglies or Stripta makaronia), pasta
 * Giosa, lamb or goat meat
 * Gournopoula (or Bouziopoula), pork
 * Hilopites
 * Kalamata olive
 * Kolokythopita (pumpkin pie)
 * Chortopita
 * Piperopita
 * Kagianas
 * Trahanas
 * Kokoras krasatos
 * Katsikaki me patates
 * Melitzanes Tsakonias (Tsakonia)
 * Syglino (pork meat) (Mani Peninsula)
 * Regali (lamb soup) (Mani)
 * Diples (dessert)
 * Galatopita (dessert)
 * Tentura drink (Patras area)

Several notable Peloponnese wines have Protected Designation of Origin (PDO) status. The Mantineia region makes a white wine made from Moschofilero, the Nemea wine region makes renowned red wines from the Agiorgitiko grape, and fortified red wine is made in the region around the city of Patras from Mavrodafni grapes.

title: Peloponnese


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User:Moshez~enwiki

I'm mostly into math, here.

For example, I was the original author of symplectic topology

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Topological vector space

In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces.

Many topological vector spaces are spaces of functions, or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions.

In this article, the scalar field of a topological vector space will be assumed to be either the complex numbers $$\Complex$$ or the real numbers $$\R,$$ unless clearly stated otherwise.

Normed spaces
Every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a topology. This is a topological vector space because:
 * 1) The vector addition map $$\cdot\, + \,\cdot\; : X \times X \to X$$ defined by $$(x, y) \mapsto x + y$$ is (jointly) continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm.
 * 2) The scalar multiplication map $$\cdot : \mathbb{K} \times X \to X$$ defined by $$(s, x) \mapsto s \cdot x,$$ where $$\mathbb{K}$$ is the underlying scalar field of $$X,$$ is (jointly) continuous. This follows from the triangle inequality and homogeneity of the norm.

Thus all Banach spaces and Hilbert spaces are examples of topological vector spaces.

Non-normed spaces
There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of holomorphic functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, and spaces of test functions and the spaces of distributions on them. These are all examples of Montel spaces. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by Kolmogorov's normability criterion.

A topological field is a topological vector space over each of its subfields.

Definition

A topological vector space (TVS) $$X$$ is a vector space over a topological field $$\mathbb{K}$$ (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that vector addition $$\cdot\, + \,\cdot\; : X \times X \to X$$ and scalar multiplication $$\cdot : \mathbb{K} \times X \to X$$ are continuous functions (where the domains of these functions are endowed with product topologies). Such a topology is called a ' or a ' on $$X.$$

Every topological vector space is also a commutative topological group under addition.

Hausdorff assumption

Many authors (for example, Walter Rudin), but not this page, require the topology on $$X$$ to be T1; it then follows that the space is Hausdorff, and even Tychonoff. A topological vector space is said to be if it is Hausdorff; importantly, "separated" does not mean separable. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below.

Category and morphisms

The category of topological vector spaces over a given topological field $$\mathbb{K}$$ is commonly denoted $$\mathrm{TVS}_\mathbb{K}$$ or $$\mathrm{TVect}_\mathbb{K}.$$ The objects are the topological vector spaces over $$\mathbb{K}$$ and the morphisms are the continuous $\mathbb{K}$-linear maps from one object to another.

A (abbreviated ), also called a, is a continuous linear map $$u : X \to Y$$ between topological vector spaces (TVSs) such that the induced map $$u : X \to \operatorname{Im} u$$ is an open mapping when $$\operatorname{Im} u := u(X),$$ which is the range or image of $$u,$$ is given the subspace topology induced by $$Y.$$

A (abbreviated ), also called a, is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.

A (abbreviated ), also called a  or an, is a bijective linear homeomorphism. Equivalently, it is a surjective TVS embedding

Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms.

A necessary condition for a vector topology

A collection $$\mathcal{N}$$ of subsets of a vector space is called if for every $$N \in \mathcal{N},$$ there exists some $$U \in \mathcal{N}$$ such that $$U + U \subseteq N.$$

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All of the above conditions are consequently a necessity for a topology to form a vector topology.

Defining topologies using neighborhoods of the origin
Since every vector topology is translation invariant (which means that for all $$x_0 \in X,$$ the map $$X \to X$$ defined by $$x \mapsto x_0 + x$$ is a homeomorphism), to define a vector topology it suffices to define a neighborhood basis (or subbasis) for it at the origin.

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In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.

Defining topologies using strings
Let $$X$$ be a vector space and let $$U_{\bull} = \left(U_i\right)_{i = 1}^{\infty}$$ be a sequence of subsets of $$X.$$ Each set in the sequence $$U_{\bull}$$ is called a  of $$U_{\bull}$$ and for every index $$i,$$ $$U_i$$ is called the $$i$$-th knot of $$U_{\bull}.$$ The set $$U_1$$ is called the beginning of $$U_{\bull}.$$ The sequence $$U_{\bull}$$ is/is a:

 *  if $$U_{i+1} + U_{i+1} \subseteq U_i$$ for every index $$i.$$
 * Balanced (resp. absorbing, closed, convex, open, symmetric, barrelled, absolutely convex/disked, etc.) if this is true of every $$U_i.$$
 *  if $$U_{\bull}$$ is summative, absorbing, and balanced.
 * ' or a ' in a TVS $$X$$ if $$U_{\bull}$$ is a string and each of its knots is a neighborhood of the origin in $$X.$$

If $$U$$is an absorbing disk in a vector space $$X$$ then the sequence defined by $$U_i := 2^{1-i} U$$ forms a string beginning with $$U_1 = U.$$ This is called the natural string of $$U$$ Moreover, if a vector space $$X$$ has countable dimension then every string contains an absolutely convex string.

Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces.

$$ A proof of the above theorem is given in the article on metrizable topological vector spaces.

If $$U_{\bull} = \left(U_i\right)_{i \in \N}$$ and $$V_{\bull} = \left(V_i\right)_{i \in \N}$$ are two collections of subsets of a vector space $$X$$ and if $$s$$ is a scalar, then by definition:

 * $$V_{\bull}$$ contains $$U_{\bull}$$: $$\ U_{\bull} \subseteq V_{\bull}$$ if and only if $$U_i \subseteq V_i$$ for every index $$i.$$
 * Set of knots: $$\ \operatorname{Knots} U_{\bull} := \left\{U_i : i \in \N\right\}.$$
 * Kernel: $\ \ker U_{\bull} := \bigcap_{i \in \N} U_i.$
 * Scalar multiple: $$\ s U_{\bull} := \left(s U_i\right)_{i \in \N}.$$
 * Sum: $$\ U_{\bull} + V_{\bull} := \left(U_i + V_i\right)_{i \in \N}.$$
 * Intersection: $$\ U_{\bull} \cap V_{\bull} := \left(U_i \cap V_i\right)_{i \in \N}.$$

If $$\mathbb{S}$$ is a collection sequences of subsets of $$X,$$ then $$\mathbb{S}$$ is said to be directed (downwards) under inclusion or simply directed downward if $$\mathbb{S}$$ is not empty and for all $$U_{\bull}, V_{\bull} \in \mathbb{S},$$ there exists some $$W_{\bull} \in \mathbb{S}$$ such that $$W_{\bull} \subseteq U_{\bull}$$ and $$W_{\bull} \subseteq V_{\bull}$$ (said differently, if and only if $$\mathbb{S}$$ is a prefilter with respect to the containment $$\,\subseteq\,$$ defined above).

Notation: Let $\operatorname{Knots} \mathbb{S} := \bigcup_{U_{\bull} \in \mathbb{S}} \operatorname{Knots} U_{\bull}$ be the set of all knots of all strings in $$\mathbb{S}.$$

Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex.

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If $$\mathbb{S}$$ is the set of all topological strings in a TVS $$(X, \tau)$$ then $$\tau_{\mathbb{S}} = \tau.$$ A Hausdorff TVS is metrizable if and only if its topology can be induced by a single topological string.

Topological structure
A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by $$-1$$). Hence, every topological vector space is an abelian topological group. Every TVS is completely regular but a TVS need not be normal.

Let $$X$$ be a topological vector space. Given a subspace $$M \subseteq X,$$ the quotient space $$X / M$$ with the usual quotient topology is a Hausdorff topological vector space if and only if $$M$$ is closed. This permits the following construction: given a topological vector space $$X$$ (that is probably not Hausdorff), form the quotient space $$X / M$$ where $$M$$ is the closure of $$\{0\}.$$ $$X / M$$ is then a Hausdorff topological vector space that can be studied instead of $$X.$$

Invariance of vector topologies
One of the most used properties of vector topologies is that every vector topology is :
 * for all $$x_0 \in X,$$ the map $$X \to X$$ defined by $$x \mapsto x_0 + x$$ is a homeomorphism, but if $$x_0 \neq 0$$ then it is not linear and so not a TVS-isomorphism.

Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if $$s \neq 0$$ then the linear map $$X \to X$$ defined by $$x \mapsto s x$$ is a homeomorphism. Using $$s = -1$$ produces the negation map $$X \to X$$ defined by $$x \mapsto - x,$$ which is consequently a linear homeomorphism and thus a TVS-isomorphism.

If $$x \in X$$ and any subset $$S \subseteq X,$$ then $$\operatorname{cl}_X (x + S) = x + \operatorname{cl}_X S$$ and moreover, if $$0 \in S$$ then $$x + S$$ is a neighborhood (resp. open neighborhood, closed neighborhood) of $$x$$ in $$X$$ if and only if the same is true of $$S$$ at the origin.

Local notions
A subset $$E$$ of a vector space $$X$$ is said to be
 * absorbing (in $$X$$): if for every $$x \in X,$$ there exists a real $$r > 0$$ such that $$c x \in E$$ for any scalar $$c$$ satisfying $$|c| \leq r.$$
 * balanced or circled: if $$t E \subseteq E$$ for every scalar $$|t| \leq 1.$$
 * convex: if $$t E + (1 - t) E \subseteq E$$ for every real $$0 \leq t \leq 1.$$
 * a disk or absolutely convex: if $$E$$ is convex and balanced.
 * symmetric: if $$- E \subseteq E,$$ or equivalently, if $$- E = E.$$

Every neighborhood of the origin is an absorbing set and contains an open balanced neighborhood of $$0$$ so every topological vector space has a local base of absorbing and balanced sets. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of $$0;$$ if the space is locally convex then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin.

Bounded subsets

A subset $$E$$ of a topological vector space $$X$$ is bounded if for every neighborhood $$V$$ of the origin there exists $$t$$ such that $$E \subseteq t V$$.

The definition of boundedness can be weakened a bit; $$E$$ is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set. Also, $$E$$ is bounded if and only if for every balanced neighborhood $$V$$ of the origin, there exists $$t$$ such that $$E \subseteq t V.$$ Moreover, when $$X$$ is locally convex, the boundedness can be characterized by seminorms: the subset $$E$$ is bounded if and only if every continuous seminorm $$p$$ is bounded on $$E.$$

Every totally bounded set is bounded. If $$M$$ is a vector subspace of a TVS $$X,$$ then a subset of $$M$$ is bounded in $$M$$ if and only if it is bounded in $$X.$$

Metrizability
$$

A TVS is pseudometrizable if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an F-seminorm. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable.

More strongly: a topological vector space is said to be normable if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of the origin.

Let $$\mathbb{K}$$ be a non-discrete locally compact topological field, for example the real or complex numbers. A Hausdorff topological vector space over $$\mathbb{K}$$ is locally compact if and only if it is finite-dimensional, that is, isomorphic to $$\mathbb{K}^n$$ for some natural number $$n.$$

Completeness and uniform structure
The canonical uniformity on a TVS $$(X, \tau)$$ is the unique translation-invariant uniformity that induces the topology $$\tau$$ on $$X.$$

Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into uniform spaces. This allows one to talk about related notions such as completeness, uniform convergence, Cauchy nets, and uniform continuity, etc., which are always assumed to be with respect to this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is Tychonoff. A subset of a TVS is compact if and only if it is complete and totally bounded (for Hausdorff TVSs, a set being totally bounded is equivalent to it being precompact). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are relatively compact).

With respect to this uniformity, a net (or sequence) $$x_{\bull} = \left(x_i\right)_{i \in I}$$ is Cauchy if and only if for every neighborhood $$V$$ of $$0,$$ there exists some index $$n$$ such that $$x_i - x_j \in V$$ whenever $$i \geq n$$ and $$j \geq n.$$

Every Cauchy sequence is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called sequentially complete; in general, it may not be complete (in the sense that all Cauchy filters converge).

The vector space operation of addition is uniformly continuous and an open map. Scalar multiplication is Cauchy continuous but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus a dense linear subspace of a complete topological vector space.

 * Every TVS has a completion and every Hausdorff TVS has a Hausdorff completion. Every TVS (even those that are Hausdorff and/or complete) has infinitely many non-isomorphic non-Hausdorff completions.
 * A compact subset of a TVS (not necessarily Hausdorff) is complete. A complete subset of a Hausdorff TVS is closed.
 * If $$C$$ is a complete subset of a TVS then any subset of $$C$$ that is closed in $$C$$ is complete.
 * A Cauchy sequence in a Hausdorff TVS $$X$$ is not necessarily relatively compact (that is, its closure in $$X$$ is not necessarily compact).
 * If a Cauchy filter in a TVS has an accumulation point $$x$$ then it converges to $$x.$$
 * If a series $\sum_{i=1}^{\infty} x_i$ converges in a TVS $$X$$ then $$x_{\bull} \to 0$$ in $$X.$$

Finest and coarsest vector topology
Let $$X$$ be a real or complex vector space.

Trivial topology

The trivial topology or indiscrete topology $$\{X, \varnothing\}$$ is always a TVS topology on any vector space $$X$$ and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on $$X$$ always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus locally compact) complete pseudometrizable seminormable locally convex topological vector space. It is Hausdorff if and only if $$\dim X = 0.$$

Finest vector topology

There exists a TVS topology $$\tau_f$$ on $$X,$$ called the  on $$X,$$ that is finer than every other TVS-topology on $$X$$ (that is, any TVS-topology on $$X$$ is necessarily a subset of $$\tau_f$$). Every linear map from $$\left(X, \tau_f\right)$$ into another TVS is necessarily continuous. If $$X$$ has an uncountable Hamel basis then $$\tau_f$$ is locally convex and  metrizable.

Cartesian products
A Cartesian product of a family of topological vector spaces, when endowed with the product topology, is a topological vector space. Consider for instance the set $$X$$ of all functions $$f: \R \to \R$$ where $$\R$$ carries its usual Euclidean topology. This set $$X$$ is a real vector space (where addition and scalar multiplication are defined pointwise, as usual) that can be identified with (and indeed, is often defined to be) the Cartesian product $$\R^\R,,$$ which carries the natural product topology. With this product topology, $$X := \R^{\R}$$ becomes a topological vector space whose topology is called The reason for this name is the following: if $$\left(f_n\right)_{n=1}^{\infty}$$ is a sequence (or more generally, a net) of elements in $$X$$ and if $$f \in X$$ then $$f_n$$ converges to $$f$$ in $$X$$ if and only if for every real number $$x,$$ $$f_n(x)$$ converges to $$f(x)$$ in $$\R.$$  This TVS is complete, Hausdorff, and locally convex but not metrizable and consequently not normable; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form $$\R f := \{r f : r \in \R\}$$ with $$f \neq 0$$).

Finite-dimensional spaces
By F. Riesz's theorem, a Hausdorff topological vector space is finite-dimensional if and only if it is locally compact, which happens if and only if it has a compact neighborhood of the origin.

Let $$\mathbb{K}$$ denote $$\R$$ or $$\Complex$$ and endow $$\mathbb{K}$$ with its usual Hausdorff normed Euclidean topology. Let $$X$$ be a vector space over $$\mathbb{K}$$ of finite dimension $$n := \dim X$$ and so that $$X$$ is vector space isomorphic to $$\mathbb{K}^n$$ (explicitly, this means that there exists a linear isomorphism between the vector spaces $$X$$ and $$\mathbb{K}^n$$). This finite-dimensional vector space $$X$$ always has a unique vector topology, which makes it TVS-isomorphic to $$\mathbb{K}^n,$$ where $$\mathbb{K}^n$$ is endowed with the usual Euclidean topology (which is the same as the product topology). This Hausdorff vector topology is also the (unique) finest vector topology on $$X.$$ $$X$$ has a unique vector topology if and only if $$\dim X = 0.$$ If $$\dim X \neq 0$$ then although $$X$$ does not have a unique vector topology, it does have a unique  vector topology.

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 * If $$\dim X = 0$$ then $$X = \{0\}$$ has exactly one vector topology: the trivial topology, which in this case (and in this case) is Hausdorff. The trivial topology on a vector space is Hausdorff if and only if the vector space has dimension $$0.$$
 * If $$\dim X = 1$$ then $$X$$ has two vector topologies: the usual Euclidean topology and the (non-Hausdorff) trivial topology.
 * Since the field $$\mathbb{K}$$ is itself a $$1$$-dimensional topological vector space over $$\mathbb{K}$$ and since it plays an important role in the definition of topological vector spaces, this dichotomy plays an important role in the definition of an absorbing set and has consequences that reverberate throughout functional analysis.
 * If $$\dim X = n \geq 2$$ then $$X$$ has distinct vector topologies:
 * Some of these topologies are now described: Every linear functional $$f$$ on $$X,$$ which is vector space isomorphic to $$\mathbb{K}^n,$$ induces a seminorm $$|f| : X \to \R$$ defined by $$|f|(x) = |f(x)|$$ where $$\ker f = \ker |f|.$$ Every seminorm induces a (pseudometrizable locally convex) vector topology on $$X$$ and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on $$X$$ that are induced by linear functionals with distinct kernels will induce distinct vector topologies on $$X.$$
 * However, while there are infinitely many vector topologies on $$X$$ when $$\dim X \geq 2,$$ there are,, only $$1 + \dim X$$ vector topologies on $$X.$$ For instance, if $$n := \dim X = 2$$ then the vector topologies on $$X$$ consist of the trivial topology, the Hausdorff Euclidean topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on $$X$$ are all TVS-isomorphic to one another.

Non-vector topologies
Discrete and cofinite topologies

If $$X$$ is a non-trivial vector space (that is, of non-zero dimension) then the discrete topology on $$X$$ (which is always metrizable) is a TVS topology because despite making addition and negation continuous (which makes it into a topological group under addition), it fails to make scalar multiplication continuous. The cofinite topology on $$X$$ (where a subset is open if and only if its complement is finite) is also a TVS topology on $$X.$$

Linear maps
A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a linear operator $$f$$ is continuous if $$f(X)$$ is bounded (as defined below) for some neighborhood $$X$$ of the origin.

A hyperplane in a topological vector space $$X$$ is either dense or closed. A linear functional $$f$$ on a topological vector space $$X$$ has either dense or closed kernel. Moreover, $$f$$ is continuous if and only if its kernel is closed.

Types
Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the closed graph theorem, the open mapping theorem, and the fact that the dual space of the space separates points in the space.

Below are some common topological vector spaces, roughly in order of increasing "niceness."

 * F-spaces are complete topological vector spaces with a translation-invariant metric. These include $L^p$ spaces for all $$p > 0.$$
 * Locally convex topological vector spaces: here each point has a local base consisting of convex sets. By a technique known as Minkowski functionals it can be shown that a space is locally convex if and only if its topology can be defined by a family of seminorms. Local convexity is the minimum requirement for "geometrical" arguments like the Hahn–Banach theorem. The $$L^p$$ spaces are locally convex (in fact, Banach spaces) for all $$p \geq 1,$$ but not for $$0 < p < 1.$$
 * Barrelled spaces: locally convex spaces where the Banach–Steinhaus theorem holds.
 * Bornological space: a locally convex space where the continuous linear operators to any locally convex space are exactly the bounded linear operators.
 * Stereotype space: a locally convex space satisfying a variant of reflexivity condition, where the dual space is endowed with the topology of uniform convergence on totally bounded sets.
 * Montel space: a barrelled space where every closed and bounded set is compact
 * Fréchet spaces: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of seminorms. Many interesting spaces of functions fall into this class -- $$C^\infty(\R)$$ is a Fréchet space under the seminorms $\|f\|_{k,\ell} = \sup_{x\in[-k,k]} |f^{(\ell)}(x)|.$ A locally convex F-space is a Fréchet space.
 * LF-spaces are limits of Fréchet spaces. ILH spaces are inverse limits of Hilbert spaces.
 * Nuclear spaces: these are locally convex spaces with the property that every bounded map from the nuclear space to an arbitrary Banach space is a nuclear operator.
 * Normed spaces and seminormed spaces: locally convex spaces where the topology can be described by a single norm or seminorm. In normed spaces a linear operator is continuous if and only if it is bounded.
 * Banach spaces: Complete normed vector spaces. Most of functional analysis is formulated for Banach spaces. This class includes the $$L^p$$ spaces with $$1\leq p \leq \infty,$$ the space $$BV$$ of functions of bounded variation, and certain spaces of measures.
 * Reflexive Banach spaces: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is reflexive is $L^1$, whose dual is $$L^{\infty}$$ but is strictly contained in the dual of $$L^{\infty}.$$
 * Hilbert spaces: these have an inner product; even though these spaces may be infinite-dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them. These include $$L^2$$ spaces, the $$L^2$$ Sobolev spaces $$W^{2,k},$$ and Hardy spaces.
 * Euclidean spaces: $$\R^n$$ or $$\Complex^n$$ with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite $$n,$$ there is only one $$n$$-dimensional topological vector space, up to isomorphism. It follows from this that any finite-dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).

Dual space
Every topological vector space has a continuous dual space—the set $$X'$$ of all continuous linear functionals, that is, continuous linear maps from the space into the base field $$\mathbb{K}.$$ A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation $$X' \to \mathbb{K}$$ is continuous. This turns the dual into a locally convex topological vector space. This topology is called the weak-* topology. This may not be the only natural topology on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach–Alaoglu theorem). Caution: Whenever $$X$$ is a non-normable locally convex space, then the pairing map $$X' \times X \to \mathbb{K}$$ is never continuous, no matter which vector space topology one chooses on $$X'.$$ A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.

Properties
For any $$S \subseteq X$$ of a TVS $$X,$$ the convex (resp. balanced, disked, closed convex, closed balanced, closed disked') hull of $$S$$ is the smallest subset of $$X$$ that has this property and contains $$S.$$ The closure (respectively, interior, convex hull, balanced hull, disked hull) of a set $$S$$ is sometimes denoted by $$\operatorname{cl}_X S$$ (respectively, $$\operatorname{Int}_X S,$$ $$\operatorname{co} S,$$ $$\operatorname{bal} S,$$ $$\operatorname{cobal} S$$).

The convex hull $$\operatorname{co} S$$ of a subset $$S$$ is equal to the set of all of elements in $$S,$$ which are finite linear combinations of the form $$t_1 s_1 + \cdots + t_n s_n$$ where $$n \geq 1$$ is an integer, $$s_1, \ldots, s_n \in S$$ and $$t_1, \ldots, t_n \in [0, 1]$$ sum to $$1.$$ The intersection of any family of convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it.

Neighborhoods and open sets
Properties of neighborhoods and open sets

Every TVS is connected and locally connected and any connected open subset of a TVS is arcwise connected. If $$S \subseteq X$$ and $$U$$ is an open subset of $$X$$ then $$S + U$$ is an open set in $$X$$ and if $$S \subseteq X$$ has non-empty interior then $$S - S$$ is a neighborhood of the origin.

The open convex subsets of a TVS $$X$$ (not necessarily Hausdorff or locally convex) are exactly those that are of the form $$z + \{x \in X : p(x) < 1\} ~=~ \{x \in X : p(x - z) < 1\}$$ for some $$z \in X$$ and some positive continuous sublinear functional $$p$$ on $$X.$$

If $$K$$ is an absorbing disk in a TVS $$X$$ and if $$p := p_K$$ is the Minkowski functional of $$K$$ then $$\operatorname{Int}_X K ~\subseteq~ \{x \in X : p(x) < 1\} ~\subseteq~ K ~\subseteq~ \{x \in X : p(x) \leq 1\} ~\subseteq~ \operatorname{cl}_X K$$ where importantly, it was assumed that $$K$$ had any topological properties nor that $$p$$ was continuous (which happens if and only if $$K$$ is a neighborhood of the origin).

Let $$\tau$$ and $$\nu$$ be two vector topologies on $$X.$$ Then $$\tau \subseteq \nu$$ if and only if whenever a net $$x_{\bull} = \left(x_i\right)_{i \in I}$$ in $$X$$ converges $$0$$ in $$(X, \nu)$$ then $$x_{\bull} \to 0$$ in $$(X, \tau).$$

Let $$\mathcal{N}$$ be a neighborhood basis of the origin in $$X,$$ let $$S \subseteq X,$$ and let $$x \in X.$$ Then $$x \in \operatorname{cl}_X S$$ if and only if there exists a net $$s_{\bull} = \left(s_N\right)_{N \in \mathcal{N}}$$ in $$S$$ (indexed by $$\mathcal{N}$$) such that $$s_{\bull} \to x$$ in $$X.$$ This shows, in particular, that it will often suffice to consider nets indexed by a neighborhood basis of the origin rather than nets on arbitrary directed sets.

If $$X$$ is a TVS that is of the second category in itself (that is, a nonmeager space) then any closed convex absorbing subset of $$X$$ is a neighborhood of the origin. This is no longer guaranteed if the set is not convex (a counter-example exists even in $$X = \R^2$$) or if $$X$$ is not of the second category in itself.

Interior

If $$R, S \subseteq X$$ and $$S$$ has non-empty interior then $$\operatorname{Int}_X S ~=~ \operatorname{Int}_X \left(\operatorname{cl}_X S\right)~ \text{ and } ~\operatorname{cl}_X S ~=~ \operatorname{cl}_X \left(\operatorname{Int}_X S\right)$$ and $$\operatorname{Int}_X (R) + \operatorname{Int}_X (S) ~\subseteq~ R + \operatorname{Int}_X S \subseteq \operatorname{Int}_X (R + S).$$

The topological interior of a disk is not empty if and only if this interior contains the origin. More generally, if $$S$$ is a balanced set with non-empty interior $$\operatorname{Int}_X S \neq \varnothing$$ in a TVS $$X$$ then $$\{0\} \cup \operatorname{Int}_X S$$ will necessarily be balanced; consequently, $$\operatorname{Int}_X S$$ will be balanced if and only if it contains the origin. For this (i.e. $$0 \in \operatorname{Int}_X S$$) to be true, it suffices for $$S$$ to also be convex (in addition to being balanced and having non-empty interior).; The conclusion $$0 \in \operatorname{Int}_X S$$ could be false if $$S$$ is not also convex; for example, in $$X := \R^2,$$ the interior of the closed and balanced set $$S := \{(x, y) : x y \geq 0\}$$ is $$\{(x, y) : x y > 0\}.$$

If $$C$$ is convex and $$0 < t \leq 1,$$ then $$t \operatorname{Int} C + (1 - t) \operatorname{cl} C ~\subseteq~ \operatorname{Int} C.$$ Explicitly, this means that if $$C$$ is a convex subset of a TVS $$X$$ (not necessarily Hausdorff or locally convex), $$y \in \operatorname{int}_X C,$$ and $$x \in \operatorname{cl}_X C$$ then the open line segment joining $$x$$ and $$y$$ belongs to the interior of $$C;$$ that is, $$\{t x + (1 - t) y : 0 < t < 1\} \subseteq \operatorname{int}_X C.$$

If $$N \subseteq X$$ is any balanced neighborhood of the origin in $$X$$ then $\operatorname{Int}_X N \subseteq B_1 N = \bigcup_{0 < |a| < 1} a N \subseteq N$ where $$B_1$$ is the set of all scalars $$a$$ such that $$|a| < 1.$$

If $$x$$ belongs to the interior of a convex set $$S \subseteq X$$ and $$y \in \operatorname{cl}_X S,$$ then the half-open line segment $$[x, y) := \{t x + (1 - t) y : 0 < t \leq 1\} \subseteq \operatorname{Int}_X \text{ if } x \neq y$$ and $$[x, x) = \varnothing \text{ if } x = y.$$ If $$N$$ is a balanced neighborhood of $$0$$ in $$X$$ and $$B_1 := \{a \in \mathbb{K} : |a| < 1\},$$ then by considering intersections of the form $$N \cap \R x$$ (which are convex symmetric neighborhoods of $$0$$ in the real TVS $$\R x$$) it follows that: $$\operatorname{Int} N = [0, 1) \operatorname{Int} N = (-1, 1) N = B_1 N,$$ and furthermore, if $$x \in \operatorname{Int} N \text{ and } r := \sup \{r > 0 : [0, r) x \subseteq N\}$$ then $$r > 1 \text{ and } [0, r) x \subseteq \operatorname{Int} N,$$ and if $$r \neq \infty$$ then $$r x \in \operatorname{cl} N \setminus \operatorname{Int} N.$$

Non-Hausdorff spaces and the closure of the origin
A topological vector space $$X$$ is Hausdorff if and only if $$\{0\}$$ is a closed subset of $$X,$$ or equivalently, if and only if $$\{0\} = \operatorname{cl}_X \{0\}.$$ Because $$\{0\}$$ is a vector subspace of $$X,$$ the same is true of its closure $$\operatorname{cl}_X \{0\},$$ which is referred to as in $$X.$$ This vector space satisfies $$\operatorname{cl}_X \{0\} = \bigcap_{N \in \mathcal{N}(0)} N$$ so that in particular, every neighborhood of the origin in $$X$$ contains the vector space $$\operatorname{cl}_X \{0\}$$ as a subset. The subspace topology on $$\operatorname{cl}_X \{0\}$$ is always the trivial topology, which in particular implies that the topological vector space $$\operatorname{cl}_X \{0\}$$ a compact space (even if its dimension is non-zero or even infinite) and consequently also a bounded subset of $$X.$$ In fact, a vector subspace of a TVS is bounded if and only if it is contained in the closure of $$\{0\}.$$ Every subset of $$\operatorname{cl}_X \{0\}$$ also carries the trivial topology and so is itself a compact, and thus also complete, subspace (see footnote for a proof). In particular, if $$X$$ is not Hausdorff then there exist subsets that are both but  in $$X$$; for instance, this will be true of any non-empty proper subset of $$\operatorname{cl}_X \{0\}.$$

If $$S \subseteq X$$ is compact, then $$\operatorname{cl}_X S = S + \operatorname{cl}_X \{0\}$$ and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are relatively compact), which is not guaranteed for arbitrary non-Hausdorff topological spaces.

For every subset $$S \subseteq X,$$ $$S + \operatorname{cl}_X \{0\} \subseteq \operatorname{cl}_X S$$ and consequently, if $$S \subseteq X$$ is open or closed in $$X$$ then $$S + \operatorname{cl}_X \{0\} = S$$ (so that this open  closed subsets $$S$$ can be described as a "tube" whose vertical side is the vector space $$\operatorname{cl}_X \{0\}$$). For any subset $$S \subseteq X$$ of this TVS $$X,$$ the following are equivalent:

 * $$S$$ is totally bounded.
 * $$S + \operatorname{cl}_X \{0\}$$ is totally bounded.
 * $$\operatorname{cl}_X S$$ is totally bounded.
 * The image if $$S$$ under the canonical quotient map $$X \to X / \operatorname{cl}_X (\{0\})$$ is totally bounded.

If $$M$$ is a vector subspace of a TVS $$X$$ then $$X / M$$ is Hausdorff if and only if $$M$$ is closed in $$X.$$ Moreover, the quotient map $$q : X \to X / \operatorname{cl}_X \{0\}$$ is always a closed map onto the (necessarily) Hausdorff TVS.

Every vector subspace of $$X$$ that is an algebraic complement of $$\operatorname{cl}_X \{0\}$$ (that is, a vector subspace $$H$$ that satisfies $$\{0\} = H \cap \operatorname{cl}_X \{0\}$$ and $$X = H + \operatorname{cl}_X \{0\}$$) is a topological complement of $$\operatorname{cl}_X \{0\}.$$ Consequently, if $$H$$ is an algebraic complement of $$\operatorname{cl}_X \{0\}$$ in $$X$$ then the addition map $$H \times \operatorname{cl}_X \{0\} \to X,$$ defined by $$(h, n) \mapsto h + n$$ is a TVS-isomorphism, where $$H$$ is necessarily Hausdorff and $$\operatorname{cl}_X \{0\}$$ has the indiscrete topology. Moreover, if $$C$$ is a Hausdorff completion of $$H$$ then $$C \times \operatorname{cl}_X \{0\}$$ is a completion of $$X \cong H \times \operatorname{cl}_X \{0\}.$$

Closed and compact sets
Compact and totally bounded sets

A subset of a TVS is compact if and only if it is complete and totally bounded. Thus, in a complete topological vector space, a closed and totally bounded subset is compact. A subset $$S$$ of a TVS $$X$$ is totally bounded if and only if $$\operatorname{cl}_X S$$ is totally bounded, if and only if its image under the canonical quotient map $$X \to X / \operatorname{cl}_X (\{0\})$$ is totally bounded.

Every relatively compact set is totally bounded and the closure of a totally bounded set is totally bounded. The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded. If $$S$$ is a subset of a TVS $$X$$ such that every sequence in $$S$$ has a cluster point in $$S$$ then $$S$$ is totally bounded.

If $$K$$ is a compact subset of a TVS $$X$$ and $$U$$ is an open subset of $$X$$ containing $$K,$$ then there exists a neighborhood $$N$$ of 0 such that $$K + N \subseteq U.$$

Closure and closed set

The closure of any convex (respectively, any balanced, any absorbing) subset of any TVS has this same property. In particular, the closure of any convex, balanced, and absorbing subset is a barrel.

The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a Hausdorff TVS is closed. The sum of a closed vector subspace and a finite-dimensional vector subspace is closed. If $$M$$ is a vector subspace of $$X$$ and $$N$$ is a closed neighborhood of the origin in $$X$$ such that $$U \cap N$$ is closed in $$X$$ then $$M$$ is closed in $$X.$$ The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed (see this footnote for examples).

If $$S \subseteq X$$ and $$a$$ is a scalar then $$a \operatorname{cl}_X S \subseteq \operatorname{cl}_X (a S),$$ where if $$X$$ is Hausdorff, $$a \neq 0, \text{ or } S = \varnothing$$ then equality holds: $$\operatorname{cl}_X (a S) = a \operatorname{cl}_X S.$$ In particular, every non-zero scalar multiple of a closed set is closed. If $$S \subseteq X$$ and if $$A$$ is a set of scalars such that neither $$\operatorname{cl} S \text{ nor } \operatorname{cl} A$$ contain zero then $$\left(\operatorname{cl} A\right) \left(\operatorname{cl}_X S\right) = \operatorname{cl}_X (A S).$$

If $$S \subseteq X \text{ and } S + S \subseteq 2 \operatorname{cl}_X S$$ then $$\operatorname{cl}_X S$$ is convex.

If $$R, S \subseteq X$$ then $$\operatorname{cl}_X (R) + \operatorname{cl}_X (S) ~\subseteq~ \operatorname{cl}_X (R + S)~ \text{ and } ~\operatorname{cl}_X \left[ \operatorname{cl}_X (R) + \operatorname{cl}_X (S) \right] ~=~ \operatorname{cl}_X (R + S)$$ and so consequently, if $$R + S$$ is closed then so is $$\operatorname{cl}_X (R) + \operatorname{cl}_X (S).$$

If $$X$$ is a real TVS and $$S \subseteq X,$$ then $$\bigcap_{r > 1} r S \subseteq \operatorname{cl}_X S$$ where the left hand side is independent of the topology on $$X;$$ moreover, if $$S$$ is a convex neighborhood of the origin then equality holds.

For any subset $$S \subseteq X,$$ $$\operatorname{cl}_X S ~=~ \bigcap_{N \in \mathcal{N}} (S + N)$$ where $$\mathcal{N}$$ is any neighborhood basis at the origin for $$X.$$ However, $$\operatorname{cl}_X U ~\supseteq~ \bigcap \{U : S \subseteq U, U \text{ is open in } X\}$$ and it is possible for this containment to be proper (for example, if $$X = \R$$ and $$S$$ is the rational numbers). It follows that $$\operatorname{cl}_X U \subseteq U + U$$ for every neighborhood $$U$$ of the origin in $$X.$$

Closed hulls

In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.

 * The closed convex hull of a set is equal to the closure of the convex hull of that set; that is, equal to $$\operatorname{cl}_X (\operatorname{co} S).$$
 * The closed balanced hull of a set is equal to the closure of the balanced hull of that set; that is, equal to $$\operatorname{cl}_X (\operatorname{bal} S).$$
 * The closed disked hull of a set is equal to the closure of the disked hull of that set; that is, equal to $$\operatorname{cl}_X (\operatorname{cobal} S).$$

If $$R, S \subseteq X$$ and the closed convex hull of one of the sets $$S$$ or $$R$$ is compact then $$\operatorname{cl}_X (\operatorname{co} (R + S)) ~=~ \operatorname{cl}_X (\operatorname{co} R) + \operatorname{cl}_X (\operatorname{co} S).$$ If $$R, S \subseteq X$$ each have a closed convex hull that is compact (that is, $$\operatorname{cl}_X (\operatorname{co} R)$$ and $$\operatorname{cl}_X (\operatorname{co} S)$$ are compact) then $$\operatorname{cl}_X (\operatorname{co} (R \cup S)) ~=~ \operatorname{co} \left[ \operatorname{cl}_X (\operatorname{co} R) \cup \operatorname{cl}_X (\operatorname{co} S) \right].$$

Hulls and compactness

In a general TVS, the closed convex hull of a compact set may to be compact. The balanced hull of a compact (respectively, totally bounded) set has that same property. The convex hull of a finite union of compact sets is again compact and convex.

Other properties
Meager, nowhere dense, and Baire

A disk in a TVS is not nowhere dense if and only if its closure is a neighborhood of the origin. A vector subspace of a TVS that is closed but not open is nowhere dense.

Suppose $$X$$ is a TVS that does not carry the indiscrete topology. Then $$X$$ is a Baire space if and only if $$X$$ has no balanced absorbing nowhere dense subset.

A TVS $$X$$ is a Baire space if and only if $$X$$ is nonmeager, which happens if and only if there does not exist a nowhere dense set $$D$$ such that $X = \bigcup_{n \in \N} n D.$ Every nonmeager locally convex TVS is a barrelled space.

Important algebraic facts and common misconceptions

If $$S \subseteq X$$ then $$2 S \subseteq S + S$$; if $$S$$ is convex then equality holds. For an example where equality does hold, let $$x$$ be non-zero and set $$S = \{- x, x\};$$ $$S = \{x, 2 x\}$$ also works.

A subset $$C$$ is convex if and only if $$(s + t) C = s C + t C$$ for all positive real $$s > 0 \text{ and } t > 0,$$ or equivalently, if and only if $$t C + (1 - t) C \subseteq C$$ for all $$0 \leq t \leq 1.$$

The convex balanced hull of a set $$S \subseteq X$$ is equal to the convex hull of the balanced hull of $$S;$$ that is, it is equal to $$\operatorname{co} (\operatorname{bal} S).$$ But in general, $$\operatorname{bal} (\operatorname{co} S) ~\subseteq~ \operatorname{cobal} S ~=~ \operatorname{co} (\operatorname{bal} S),$$ where the inclusion might be strict since the balanced hull of a convex set need not be convex (counter-examples exist even in $$\R^2$$).

If $$R, S \subseteq X$$ and $$a$$ is a scalar then $$a(R + S) = aR + a S,~ \text{ and } ~\operatorname{co} (R + S) = \operatorname{co} R + \operatorname{co} S,~ \text{ and } ~\operatorname{co} (a S) = a \operatorname{co} S.$$ If $$R, S \subseteq X$$ are convex non-empty disjoint sets and $$x \not\in R \cup S,$$ then $$S \cap \operatorname{co} (R \cup \{x\}) = \varnothing $$ or $$R \cap \operatorname{co} (S \cup \{x\}) = \varnothing.$$

In any non-trivial vector space $$X,$$ there exist two disjoint non-empty convex subsets whose union is $$X.$$

Other properties

Every TVS topology can be generated by a of F-seminorms.

If $$P(x)$$ is some unary predicate (a true or false statement dependent on $$x \in X$$) then for any $$z \in X,$$ $$z + \{x \in X : P(x)\} = \{x \in X : P(x - z)\}.$$ So for example, if $$P(x)$$ denotes "$$\|x\| < 1$$" then for any $$z \in X,$$ $$z + \{x \in X : \|x\| < 1\} = \{x \in X : \|x - z\| < 1\}.$$ Similarly, if $$s \neq 0$$ is a scalar then $$s \{x \in X : P(x)\} = \left\{x \in X : P\left(\tfrac{1}{s} x\right)\right\}.$$ The elements $$x \in X$$ of these sets must range over a vector space (that is, over $$X$$) rather than not just a subset or else these equalities are no longer guaranteed; similarly, $$z$$ must belong to this vector space (that is, $$z \in X$$).

Properties preserved by set operators

 * The balanced hull of a compact (respectively, totally bounded, open) set has that same property.
 * The (Minkowski) sum of two compact (respectively, bounded, balanced, convex) sets has that same property. But the sum of two closed sets need be closed.
 * The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need be closed. And the convex hull of a bounded set need  be bounded.

The following table, the color of each cell indicates whether or not a given property of subsets of $$X$$ (indicated by the column name, "convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red.

So for instance, since the union of two absorbing sets is again absorbing, the cell in row "$$R \cup S$$" and column "Absorbing" is colored green. But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections (of at least 1 set)" and column "Absorbing" is colored red. If a cell is not colored then that information has yet to be filled in.

title: Topological vector space


text:
Where Mathematics Comes From

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (hereinafter WMCF) is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. Published in 2000, WMCF seeks to found a cognitive science of mathematics, a theory of embodied mathematics based on conceptual metaphor.

WMCF definition of mathematics
Mathematics makes up that part of the human conceptual system that is special in the following way:
 * It is precise, consistent, stable across time and human communities, symbolizable, calculable, generalizable, universally available, consistent within each of its subject matters, and effective as a general tool for description, explanation, and prediction in a vast number of everyday activities, [ranging from] sports, to building, business, technology, and science. - WMCF, pp. 50, 377

Nikolay Lobachevsky said "There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world." A common type of conceptual blending process would seem to apply to the entire mathematical procession.

Human cognition and mathematics

Lakoff and Núñez's avowed purpose is to begin laying the foundations for a truly scientific understanding of mathematics, one grounded in processes common to all human cognition. They find that four distinct but related processes metaphorically structure basic arithmetic: object collection, object construction, using a measuring stick, and moving along a path.

WMCF builds on earlier books by Lakoff (1987) and Lakoff and Johnson (1980, 1999), which analyze such concepts of metaphor and image schemata from second-generation cognitive science. Some of the concepts in these earlier books, such as the interesting technical ideas in Lakoff (1987), are absent from WMCF.

Lakoff and Núñez hold that mathematics results from the human cognitive apparatus and must therefore be understood in cognitive terms. WMCF advocates (and includes some examples of) a cognitive idea analysis of mathematics which analyzes mathematical ideas in terms of the human experiences, metaphors, generalizations, and other cognitive mechanisms giving rise to them. A standard mathematical education does not develop such idea analysis techniques because it does not pursue considerations of A) what structures of the mind allow it to do mathematics or B) the philosophy of mathematics.

Lakoff and Núñez start by reviewing the psychological literature, concluding that human beings appear to have an innate ability, called subitizing, to count, add, and subtract up to about 4 or 5. They document this conclusion by reviewing the literature, published in recent decades, describing experiments with infant subjects. For example, infants quickly become excited or curious when presented with "impossible" situations, such as having three toys appear when only two were initially present.

The authors argue that mathematics goes far beyond this very elementary level due to a large number of metaphorical constructions. For example, the Pythagorean position that all is number, and the associated crisis of confidence that came about with the discovery of the irrationality of the square root of two, arises solely from a metaphorical relation between the length of the diagonal of a square, and the possible numbers of objects.

Much of WMCF deals with the important concepts of infinity and of limit processes, seeking to explain how finite humans living in a finite world could ultimately conceive of the actual infinite. Thus much of WMCF is, in effect, a study of the epistemological foundations of the calculus. Lakoff and Núñez conclude that while the potential infinite is not metaphorical, the actual infinite is. Moreover, they deem all manifestations of actual infinity to be instances of what they call the "Basic Metaphor of Infinity", as represented by the ever-increasing sequence 1, 2, 3, ...

WMCF emphatically rejects the Platonistic philosophy of mathematics. They emphasize that all we know and can ever know is human mathematics, the mathematics arising from the human intellect. The question of whether there is a "transcendent" mathematics independent of human thought is a meaningless question, like asking if colors are transcendent of human thought—colors are only varying wavelengths of light, it is our interpretation of physical stimuli that make them colors.

WMCF (p. 81) likewise criticizes the emphasis mathematicians place on the concept of closure. Lakoff and Núñez argue that the expectation of closure is an artifact of the human mind's ability to relate fundamentally different concepts via metaphor.

WMCF concerns itself mainly with proposing and establishing an alternative view of mathematics, one grounding the field in the realities of human biology and experience. It is not a work of technical mathematics or philosophy. Lakoff and Núñez are not the first to argue that conventional approaches to the philosophy of mathematics are flawed. For example, they do not seem all that familiar with the content of Davis and Hersh (1981), even though the book warmly acknowledges Hersh's support.

Lakoff and Núñez cite Saunders Mac Lane (the inventor, with Samuel Eilenberg, of category theory) in support of their position. Mathematics, Form and Function (1986), an overview of mathematics intended for philosophers, proposes that mathematical concepts are ultimately grounded in ordinary human activities, mostly interactions with the physical world.

Educators have taken some interest in what WMCF suggests about how mathematics is learned, and why students find some elementary concepts more difficult than others.

However, even from an educational perspective, WMCF is still problematic. From the conceptual metaphor theory's point of view, metaphors reside in a different realm, the abstract, from that of 'real world', the concrete. In other words, despite their claim of mathematics being human,  established mathematical knowledge — which is what we learn in school — is assumed to be and treated as abstract, completely detached from its physical origin. It cannot account for the way learners could access to such knowledge.

WMCF is also criticized for its monist approach. First, it ignores the fact that the sensori-motor experience upon which our linguistic structure — thus, mathematics — is assumed to be based may vary across cultures and situations. Second, the mathematics WMCF is concerned with is "almost entirely... standard utterances in textbooks and curricula", which is the most-well established body of knowledge. It is negligent of the dynamic and diverse nature of the history of mathematics.

WMCF's logo-centric approach is another target for critics. While it is predominantly interested in the association between language and mathematics, it does not account for how non-linguistic factors contribute to the emergence of mathematical ideas (e.g. See Radford, 2009; Rotman, 2008 ).

Examples of mathematical metaphors
Conceptual metaphors described in WMCF, in addition to the Basic Metaphor of Infinity, include: Mathematical reasoning requires variables ranging over some universe of discourse, so that we can reason about generalities rather than merely about particulars. WMCF argues that reasoning with such variables implicitly relies on what it terms the Fundamental Metonymy of Algebra.
 * Arithmetic is motion along a path, object collection/construction;
 * Change is motion;
 * Sets are containers, objects;
 * Continuity is gapless;
 * Mathematical systems have an "essence," namely their axiomatic algebraic structure;
 * Functions are sets of ordered pairs, curves in the Cartesian plane;
 * Geometric figures are objects in space;
 * Logical independence is geometric orthogonality;
 * Numbers are sets, object collections, physical segments, points on a line;
 * Recurrence is circular.

Example of metaphorical ambiguity
WMCF (p. 151) includes the following example of what the authors term "metaphorical ambiguity." Take the set $$A = \{\{\emptyset\},\{\emptyset, \{\emptyset\}\}\}.$$ Then recall two bits of standard terminology from elementary set theory: By (1), A is the set {1,2}. But (1) and (2) together say that A is also the ordered pair (0,1). Both statements cannot be correct; the ordered pair (0,1) and the unordered pair {1,2} are fully distinct concepts. Lakoff and Johnson (1999) term this situation "metaphorically ambiguous." This simple example calls into question any Platonistic foundations for mathematics.
 * 1) The recursive construction of the ordinal natural numbers, whereby 0 is $$\empty$$, and $$n+1$$ is $$n \cup \{n\}.$$
 * 2) The ordered pair (a,b), defined as $$\{\{a\},\{a,b\}\}.$$

While (1) and (2) above are admittedly canonical, especially within the consensus set theory known as the Zermelo–Fraenkel axiomatization, WMCF does not let on that they are but one of several definitions that have been proposed since the dawning of set theory. For example, Frege, Principia Mathematica, and New Foundations (a body of axiomatic set theory begun by Quine in 1937) define cardinals and ordinals as equivalence classes under the relations of equinumerosity and similarity, so that this conundrum does not arise. In Quinian set theory, A is simply an instance of the number 2. For technical reasons, defining the ordered pair as in (2) above is awkward in Quinian set theory. Two solutions have been proposed:
 * A variant set-theoretic definition of the ordered pair more complicated than the usual one;
 * Taking ordered pairs as primitive.

The Romance of Mathematics
The "Romance of Mathematics" is WMCF's light-hearted term for a perennial philosophical viewpoint about mathematics which the authors describe and then dismiss as an intellectual myth:
 * Mathematics is transcendent, namely it exists independently of human beings, and structures our actual physical universe and any possible universe. Mathematics is the language of nature, and is the primary conceptual structure we would have in common with extraterrestrial aliens, if any such there be.
 * Mathematical proof is the gateway to a realm of transcendent truth.
 * Reasoning is logic, and logic is essentially mathematical. Hence mathematics structures all possible reasoning.
 * Because mathematics exists independently of human beings, and reasoning is essentially mathematical, reason itself is disembodied. Therefore, artificial intelligence is possible, at least in principle.

It is very much an open question whether WMCF will eventually prove to be the start of a new school in the philosophy of mathematics. Hence the main value of WMCF so far may be a critical one: its critique of Platonism and romanticism in mathematics.

Critical response
Many working mathematicians resist the approach and conclusions of Lakoff and Núñez. Reviews of WMCF by mathematicians in professional journals, while often respectful of its focus on conceptual strategies and metaphors as paths for understanding mathematics, have taken exception to some of the WMCF's philosophical arguments on the grounds that mathematical statements have lasting 'objective' meanings. For example, Fermat's Last Theorem means exactly what it meant when Fermat initially proposed it 1664. Other reviewers have pointed out that multiple conceptual strategies can be employed in connection with the same mathematically defined term, often by the same person (a point that is compatible with the view that we routinely understand the 'same' concept with different metaphors). The metaphor and the conceptual strategy are not the same as the formal definition which mathematicians employ. However, WMCF points out that formal definitions are built using words and symbols that have meaning only in terms of human experience.

Critiques of WMCF include the humorous:

and the physically informed:

Lakoff made his reputation by linking linguistics to cognitive science and the analysis of metaphor. Núñez, educated in Switzerland, is a product of Jean Piaget's school of cognitive psychology as a basis for logic and mathematics. Núñez has thought much about the foundations of real analysis, the real and complex numbers, and the Basic Metaphor of Infinity. These topics, however, worthy though they be, form part of the superstructure of mathematics. Indeed, the authors do pay a fair bit of attention early on to logic, Boolean algebra and the Zermelo–Fraenkel axioms, even lingering a bit over group theory. But neither author is well-trained in logic, the philosophy of set theory, the axiomatic method, metamathematics, and model theory. Nor does WMCF say enough about the derivation of number systems (the Peano axioms go unmentioned), abstract algebra, equivalence and order relations, mereology, topology, and geometry.

Lakoff and Núñez tend to dismiss the negative opinions mathematicians have expressed about WMCF, because their critics do not appreciate the insights of cognitive science. Lakoff and Núñez maintain that their argument can only be understood using the discoveries of recent decades about the way human brains process language and meaning. They argue that any arguments or criticisms that are not grounded in this understanding cannot address the content of the book.

It has been pointed out that it is not at all clear that WMCF establishes that the claim "intelligent alien life would have mathematical ability" is a myth. To do this, it would be required to show that intelligence and mathematical ability are separable, and this has not been done. On Earth, intelligence and mathematical ability seem to go hand in hand in all life-forms, as pointed out by Keith Devlin among others. The authors of WMCF have not explained how this situation would (or even could) be different anywhere else.

Lakoff and Núñez also appear not to appreciate the extent to which intuitionists and constructivists have anticipated their attack on the Romance of (Platonic) Mathematics. Brouwer, the founder of the intuitionist/constructivist point of view, in his dissertation On the Foundation of Mathematics, argued that mathematics was a mental construction, a free creation of the mind and totally independent of logic and language. He goes on to upbraid the formalists for building verbal structures that are studied without intuitive interpretation. Symbolic language should not be confused with mathematics; it reflects, but does not contain, mathematical reality.

Summing up
WMCF (pp. 378–79) concludes with some key points, a number of which follow. Mathematics arises from our bodies and brains, our everyday experiences, and the concerns of human societies and cultures. It is:
 * The result of normal adult cognitive capacities, in particular the capacity for conceptual metaphor, and as such is a human universal. The ability to construct conceptual metaphors is neurologically based, and enables humans to reason about one domain using the language and concepts of another domain. Conceptual metaphor is both what enabled mathematics to grow out of everyday activities, and what enables mathematics to grow by a continual process of analogy and abstraction;
 * Symbolic, thereby enormously facilitating precise calculation;
 * Not transcendent, but the result of human evolution and culture, to which it owes its effectiveness. During experience of the world a connection to mathematical ideas is going on within the human mind;
 * A system of human concepts making extraordinary use of the ordinary tools of human cognition;
 * An open-ended creation of human beings, who remain responsible for maintaining and extending it;
 * One of the greatest products of the collective human imagination, and a magnificent example of the beauty, richness, complexity, diversity, and importance of human ideas.

The cognitive approach to formal systems, as described and implemented in WMCF, need not be confined to mathematics, but should also prove fruitful when applied to formal logic, and to formal philosophy such as Edward Zalta's theory of abstract objects. Lakoff and Johnson (1999) fruitfully employ the cognitive approach to rethink a good deal of the philosophy of mind, epistemology, metaphysics, and the history of ideas.

title: Where Mathematics Comes From