No of course he isn't, no fish is happy in a bowl.
Bowls are for soup, not for fish. A Betta should be in at least a 2.5 gallon tank with a filter and heater.
EDIT: The first answerer is right about the larger tank, but wrong about other Bettas. Male Bettas cannot be housed with other Bettas or else they will fight to death. However, if the tank is big enough, they can be housed with other peaceful fish that don't have any long fins he can nip.