Actually, the founder of Normandy was William's Great-Great-Great-Grandfather
Rollo. Rollo was one of several viking pirates who sailed up the Seine River to loot Paris and the surrounding French countryside.
Since Normandy was lightly populated at the time (885 a.d.), the French made a deal with Rollo that they would give his group of pirates and their families the province of Normandy if they would
- learn French.
- convert to Christianity.
- protect the French interior from other raiders.
- Pledge their allegiance to the French king.
The Normans, thinking the climate far more agreeable than their native Norway, accepted the offer. Of course, there was nothing in the treaty with the French which prevented them from raiding the British Isles and elsewhere. The
Norman conquests ranged as far south as the Mediterranean including modern day Sicily and Israel.