Because William conquered England by defeating Harold in the Battle of Hastings in 1066. He was crowned on Christmas Day that year in Westminster Abbey.
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The general public at the time I believe, because it was well known that his father and mother weren't married.
William, Duke of Normandy, who become known as William the Conquerer after he became King of England.
The Domesday Book was a major survey of people and property in England, carried out in 1086. It was ordered by King William I, also known as William the conqueror, and its main purpose was raising taxes.
William the Conqueror
Last names (surnames) became more common after the Domesday Book (1087) which William instituted. However, monarchs were and are seldom known by a last name. Had William been given a patronymic name, he might have been known as William Robertson or William Robinson, for his father Robert I. Or the dynastic name William Norman could have been used.
William the Conqueror