Why the Allies did not invade northern Norway instead of Italy

Why did the Allies not invade northern Norway instead of Italy and use Sweden and northern bases against Germany?

Northern Norway was an exceptionally difficult theater, and commando threats already pinned a large German garrison there without forcing the Allies to fight it.

The question proposes invading northern Norway, drawing Sweden to the Allied side, and gaining submarine and air bases close to Germany. The terrain and supply problem would be even worse than advancing north through Italy. Germany had already filled Norway with troops because Allied commando activity kept the possibility of invasion alive. That result served Allied strategy: large German forces remained far from the decisive fronts while having little useful work to do. An actual invasion would convert a successful economy-of-force deception into a costly campaign against a prepared garrison. Fixing enemy troops in place can be more valuable than attacking the position they defend.