Lucien Chardon, the son of a lower middle-class father and an impoverished mother of remote aristocratic descent, is the pivotal figure of the entire work. Living at Angoulême, he is impoverished, impatient, handsome and ambitious. His widowed mother, his sister Ève and his best friend, David Séchard, do nothing to lessen his high opinion of his own talents, for it is an opinion they share.
Even as Part I of Illusions perdues, Les Deux poètes (The Two Poets), begins, Lucien has already written a historical novel and a sonnet sequence, whereas David is a scientist. But both, according to Balzac, are "poets" in that they creatively seek truth. Theirs is a fraternity of poetic aspiration, whether as scientist or writer: thus, even before David marries Ève, the two young men are spiritual brothers.
Lucien is introduced into the drawing-room of the leading figure of Angoulême high society, Mme de Bargeton, who rapidly becomes infatuated with him. It is not long before the pair flee to Paris where Lucien adopts his maternal patronymic of de Rubempré and hopes to make his mark as a poet. Mme de Bargeton, on the other hand, recognises her mésalliance and, though remaining in Paris, severs all ties with Lucien, abandoning him to a life of destitution.
In Part II, Un Grand homme de province à Paris, Lucien is contrasted both with the journalist Lousteau and the high-minded writer Daniel d’Arthez. Jilted by Mme de Bargeton for the adventurer Sixte du Châtelet, he moves in a social circle of high-class actress-prostitutes and their journalist lovers: soon he becomes the lover of Coralie. As a literary journalist he prostitutes his talent. But he still harbours the ambition of belonging to high society and longs to assume by royal warrant the surname and coat of arms of the de Rubemprés. He therefore switches his allegiance from the liberal opposition press to the one or two royalist newspapers that support the government. This act of betrayal earns him the implacable hatred of his erstwhile journalist colleagues, who destroy Coralie’s theatrical reputation. In the depths of his despair he forges his brother-in-law’s name on three promissory notes. This is his ultimate betrayal of his integrity as a person. After Coralie’s death he returns in disgrace to Angoulême, stowed away behind the Châtelets’ carriage: Mme de Bargeton has just married du Châtelet, who has been appointed prefect of that region.
Meanwhile, at Angoulême David Séchard is betrayed on all sides but is supported by his loving wife. He invents a new and cheaper method of paper production: thus, at a thematic level, the commercialization of paper-manufacturing processes is very closely interwoven with the commercialization of literature. Lucien’s forgery of his brother-in-law’s signature almost bankrupts David, who has to sell the secret of his invention to business rivals. Lucien is about to commit suicide when he is approached by a sham Jesuit priest, the Abbé Carlos Herrera: this, in another guise, is the escaped convict Vautrin whom Balzac had already presented in Le Père Goriot. Herrera takes Lucien under his protection and they drive off to Paris, there to begin a fresh assault on the capital.

Question: By marrying Eve, David becomes Lucien's what?
Answer: Brother in law