Airspace, yes. Space space, no -- Jurisdiction doesn't extend above the atmosphere. Rather, objects in space are governed by and the legal responsibility of the country from which they launched, per the Outer Space Treaty.
The reason for this, by the way, dates back to a bit of cold war strategising which pre-dated even Sputnik. Both the Americans and Soviets decided that it was more important to be able to know what the other was doing than to extend their territorial claims up to orbit. This helped to keep the doctrine of Mutually Assured Destruction in a relatively static lock; otherwise it would have been far mor dynamic and dangerous. Therefore the US did not protest when Sputnik flew overhead, and the USSR did not object to American satellites; in 1967 this was formalised by the Outer Space Treaty:
The reason for this, by the way, dates back to a bit of cold war strategising which pre-dated even Sputnik. Both the Americans and Soviets decided that it was more important to be able to know what the other was doing than to extend their territorial claims up to orbit. This helped to keep the doctrine of Mutually Assured Destruction in a relatively static lock; otherwise it would have been far mor dynamic and dangerous. Therefore the US did not protest when Sputnik flew overhead, and the USSR did not object to American satellites; in 1967 this was formalised by the Outer Space Treaty:
http://en.wikipedia.org/wiki/Outer_Space_Treaty