Primavera, is a large panel painting in tempera paint by the Italian Renaissance painter Sandro Botticelli made in the late 1470s or early 1480s. It has been described as "one of the most written about, most controversial paintings in the world", "one of the most popular paintings in Western art"; the painting depicts a group of figures from classical mythology in a garden, but no story has been found that brings this particular group together. Most critics agree that the painting is an allegory based on the lush growth of Spring, but accounts of any precise meaning vary, though many involve the Renaissance Neoplatonism which fascinated intellectual circles in Florence; the subject was first described as Primavera by the art historian Giorgio Vasari who saw it at Villa Castello, just outside Florence, by 1550. Although the two are now known not to be a pair, the painting is discussed with Botticelli's other large mythological painting, The Birth of Venus in the Uffizi, they are among the most famous paintings in the world, icons of the Italian Renaissance.

As depictions of subjects from classical mythology on a large scale, they were unprecedented in Western art since classical antiquity. The history of the painting is not known, it draws from a number of classical and Renaissance literary sources, including the works of the Ancient Roman poet Ovid and, less Lucretius, may allude to a poem by Poliziano, the Medici house poet who may have helped Botticelli devise the composition. Since 1919 the painting has been part of the collection of the Uffizi Gallery in Italy; the painting features two male, along with a cupid, in an orange grove. The movement of the composition is from right to left, so following that direction the standard identification of the figures is: at far right "Zephyrus, the biting wind of March and possesses the nymph Chloris, whom he marries and transforms into a deity. Chloris the nymph overlaps the goddess she transforms into. In the centre and somewhat set back from the other figures stands Venus, a red-draped woman in blue.

Like the flower-gatherer, she returns the viewer's gaze. The trees behind her form a broken arch to draw the eye. In the air above her a blindfolded Cupid aims his bow to the left. On the left of the painting the Three Graces, a group of three females in diaphanous white, join hands in a dance. At the extreme left Mercury, clothed in red with a sword and a helmet, raises his caduceus or wooden rod towards some wispy gray clouds; the interactions between the figures are enigmatic. Zephyrus and Chloris are looking at each other. Flora and Venus look out at the viewer, the Cupid is blindfolded, Mercury has turned his back on the others, looks up at the clouds; the central Grace looks towards him. Flora's smile was unusual in painting at this date; the pastoral scenery is elaborate. There are 500 identified plant species depicted in the painting, with about 190 different flowers, of which at least 130 can be identified; the overall appearance, size, of the painting is similar to that of the millefleur Flemish tapestries that were popular decorations for palaces at the time.

These tapestries had not caught up by the 1480s with the artistic developments of the Italian Renaissance, the composition of the painting has aspects that belong to this still Gothic style. The figures are spread in a rough line across the front of the picture space, "set side by side like pearls on a string", it is now known that in the setting for which the painting was designed the bottom was about at eye level, or above it explaining "the rising plane" on which the figures stand. The feet of Venus are higher than those of the others, showing she is behind them, but she is at the same scale, if not larger, than the other figures. Overlapping of other figures by Mercury's sword and Chloris' hands shows that they stand in front of the left Grace and Flora which might not be obvious otherwise, for example from their feet, it has been argued that the flowers do not grow smaller to the rear of the picture space a feature of the millefleur tapestries. The costumes of the figures are versions of the dress of contemporary Florence, though the sort of "quasi-theatrical costumes designed for masquerades of the sort that Vasari wrote were invented by Lorenzo de' Medici for civic festivals and tournaments."

The lack of an obvious narrative may relate to the world of pageants and tableau vivants as well as static Gothic allegories. Various interpretations of the figures have been set forth, but it is agreed that at least at one level the painting is "an elaborate mythological allegory of the burgeoning fertility of the world." It is thought that Botticelli had help devising the composition of the painting and whatever meanings it was intended to contain, as it appears that the painting reflects a deep knowledge of classical literature and philosophy that Botticelli is unlikely to have possessed. Poliziano is thought to have been involved in this, though Marsilio Ficino, another member of Lorenzo de' Medici's circle and a key figure in Renaissance Neoplatonism, has often been mentioned. One aspect of the painting is a depiction of the progress of the season of spring, reading from right to left

In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, it shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, has numerous uses in convex geometry; the theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C of continuous functions on an interval was proved earlier by Eduard Helly, a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz; the most general formulation of the theorem needs some preparation. Given a real vector space V, a function f: V → R is called sublinear if Positive homogeneity: f = γ f for all γ ∈ R+, x ∈ V, Subadditivity: f ≤ f + f for all x, y ∈ V.

Every seminorm on V is sublinear. Other sublinear functions can be useful as well Minkowski functionals of convex sets. Hahn–Banach theorem. If p: V → R is a sublinear function, φ: U → R is a linear functional on a linear subspace U ⊆ V, dominated by p on U, i.e. φ ≤ p ∀ x ∈ U there exists a linear extension ψ: V → R of φ to the whole space V, i.e. there exists a linear functional ψ such that ψ = φ ∀ x ∈ U, ψ ≤ p ∀ x ∈ V. Hahn–Banach theorem. Set K = R or C and let V be a K-vector space with a seminorm p: V → R. If φ: U → K is a K-linear functional on a K-linear subspace U of V, dominated by p on U in absolute value, | φ | ≤ p ∀ x ∈ U there exists a linear extension ψ: V → K of φ to the whole space V, i.e. there exists a K-linear functional ψ such that ψ = φ ∀ x ∈ U, | ψ | ≤ p ∀ x ∈ V. In the complex case of the alternate version, the C-linearity assumptions demand, in addition to the assumptions for the real case, that for every vector x ∈ U, we have ix ∈ U and φ = iφ; the extension ψ is in general not uniquely specified by φ and the proof gives no explicit method as to how to find ψ.

The usual proof for the case of an infinite dimensional space V uses Zorn's lemma or, the axiom of choice. It is now known that the ultrafilter lemma, weaker than the axiom of choice, is strong enough, it is possible to relax the subadditivity condition on p, requiring only that: p ≤ | a | p + | b | p, x, y ∈ V, | a | + | b | ≤ 1. It is further possible to relax the positive homogeneity and the subadditivity conditions on p, requiring only that p is convex; this reveals the intimate connection between the Hahn -- convexity. The Mizar project has formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file; the theorem has several important consequences, some of which are sometimes called "Hahn–Banach theorem": If V is a normed vector space with linear subspace U and if φ: U → K is continuous and linear there exists an extension ψ: V → K of φ, continuous and linear and which has the same operator norm as φ. In other words, in the category of normed vector spaces, the space K is an injective object.

If V is a normed vector space with linear subspace U and if z is an element of V not in the closure of U there exists a continuous linear map ψ: V → K with ψ = 0 for all x in U, ψ = 1, ||ψ|| = dist−1. In particular, if V is a normed vector space and z is an element of V there exists a continuous linear map ψ: V → K such that ψ = ||z|| and ||ψ|| ≤ 1; this implies that the natural injection J from a normed space V into its double dual V′′ is isometric. Hahn–Banach separa

"The Brothers Jones" is the fifteenth episode of the fifth season of the American fantasy drama series Once Upon a Time, which aired on March 27, 2016. In this episode, Emma is suspicious of Liam. In flashbacks, Hades makes a deal with Liam; the Jewel of the Realm ship appears in the red-tinted forest. The Enchanted Forest events from take place after the first flashback scene of "Swan Song" and before "Good Form"; the Underworld events take place after "Devil's Due". In the early years of the Enchanted Forest and Killian are miserable and under servitude after their father sold them into slavery. Liam is hoping. Killian becomes drunk and gambles away their prospects. Although Liam is still able to join, he vows to not leave without his brother; some time their ship crosses paths with a hurricane en route to a sought-after treasure described as "The Eye of the Storm." Knowing the dangers of venturing into the hurricane, Liam holds a successful mutiny against the crew's Captain Silver. Soon after the mutiny, Hades appears in the Captain's quarters and makes a deal with Liam: Hades will to take the souls of the men on the ship besides those of Liam and his brother in return for the Eye of the Storm.

Liam sails into the hurricane that kills the crew but spares the Jones brothers. Liam gives the treasure to the Royal Navy and he and Killian are declared heroes, receive positions in the Navy, given a ship: The Jewel of the Realm, which would be known as the Jolly Roger. In the Underworld and Henry are searching for the Author's quill, they split up and Henry notices a glimmering light, which leads him to the now-deceased Apprentice. The Apprentice explains that Henry is his unfinished business, that he won't be able to move on until he knows that Henry will make the right decision. At Henry's behest, the Apprentice tells him that the quill is located in the Sorcerer's mansion, but warns Henry against using its power to resurrect Cruella. Meanwhile, Emma uses her magic to heal Hook, but when she tried to kiss him Hook turns away, blaming himself for becoming dark; the two are interrupted by Killian's dead brother Liam. Liam explains rumors of a book with the power to defeat Hades. Emma assumes it to be the Underworld equivalent of the "Once Upon a Time" book.

Everyone come up empty. Concerned that Emma is not good enough for his brother, Liam tells Emma that Hook and her should break up. Moments Henry returns and suggests they look for the book in the Sorcerer’s mansion. However, the key is located in the Sheriff's office, occupied by James. After they leave, Hades reveals that he knows of their plans. Hades blackmails Liam to destroy the Storybook pages concerning him, threatening to reveal to Liam's unfinished business to his brother. At the Sheriff's office and David find the key, only to encounter Cruella. While Mary Margaret hides, David pretends to be his brother James. At the mansion, Regina and Liam look for the storybook while Henry searches for the quill. Unknown to everyone, Liam rips out the pages about Hades. At the same time, Henry rediscovers the quill. Meanwhile, David turns down Cruella's advances and learns that James has resented his brother for staying with their mother during their separate childhoods while James lived with his new father, the king.

The others notice its missing pages. While they keep looking, Liam tosses the pages outside the Mansion. Emma suspects Liam of lying and tells Hook, who refuses to believe her, suggests that they break up after escaping the Underworld. After Emma storms off, Hook notices that Liam has ink on his fingers, Captain Silver and the deceased crew appear, they reveal that Liam sold their lives for the Eye of the Storm. Liam and Killian are taken to the edge of the Underworld where they would be sent to an worse afterlife. However, Hades appears, blows Captain Silver into the Phlegethon, the River of Fire, offers Liam a chance to escape, pleased that he destroyed the pages. However, Liam refuses to go with Hades. Hook is unable to. Liam falls into the fiery abyss. Despite this, a boat appears with Liam, ready to enter a better afterlife. Seeing this, Hades disappears; the deceased crew is able to leave. Liam tells Hook he was wrong about Emma. Hook explains what has happened, revealing he now wants to return with her.

Henry tells David that he has found the quill, but admits that he plans to use the quill only in a positive way, starting with re-writing the missing Hades story that Liam ripped out. Meanwhile, Hades retrieves the missing pages from the Cocytus River; the pages show. Emilie de Ravin, Rebecca Mader, Sean Maguire, Robert Carlyle are not featured in this episode; the episode was met with mixed reviews. In a review from Rickey.org, Nick Roman said, "“The Brothers Jones” is a strong episode for characterization on Once Upon A Time, since it illustrates how several characters have both matured and acquired a much-needed sense of self-awareness. I’ve liked other episodes better this season, but I would argue this was one of the best when it comes to pure character study."Andrea Towers of Entertainment Weekly gave it a good review, noting t