The Spitsbergen Treaty was signed in Paris on 9 February 1920, during the Versailles negotiations after World War I. In this treaty, international diplomacy recognized Norwegian sovereignty (the Norwegian administration went in effect by 1925) and other principles relating to Svalbard. This includes:[9]

Svalbard is part of Norway: Svalbard is completely controlled by and forms part of the Kingdom of Norway. However, Norway's power over Svalbard is restricted by the limitations listed below:

Taxation: Taxes are allowed to be collected, but only enough to support Svalbard and the Svalbard government. This results in lower taxes than mainland Norway and the exclusion of any taxes on Svalbard supporting mainland Norway directly. Svalbard's revenues and expenses are separately budgeted from mainland Norway.

Environmental conservation: Norway must respect and preserve the Svalbard environment.

Non-discrimination: All citizens and all companies of every nation under the treaty are allowed to become residents and to have access to Svalbard including the right to fish, hunt or undertake any kind of maritime, industrial, mining or trade activity. The residents of Svalbard must follow Norwegian law, though Norwegian authority cannot discriminate against or favour any residents of any given nationality. Russia has exercised this right of residency and in 2022 announced new investment plans to support its presence at Barentsburg and Pyramiden.[18]

Military restrictions: Article 9 prohibits naval bases and fortifications and also the use of Svalbard for war-like purposes. Norway interprets this provision as still permitting access to the territory by the Norwegian Armed Forces to exercise sovereignty and protect the environment. This largely encompasses visits by Norwegian military forces, particularly Norwegian Coast Guard vessels, to the territory. However, certain other signatory states, particularly Russia, dispute this interpretation. Notably, while Norway maintains that it has exclusive rights under the United Nations Convention on the Law of the Sea to the continental shelf around Svalbard, Russia rejects this position. Russia also objects to Norwegian or NATO military activities around Svalbard which, Russia claims, undermine the treaty’s military restrictions.[19]

(http://www.autoadmit.com/thread.php?thread_id=5868609&forum_id=2/#49897788)

Also, and this isn’t mentioned here, but if you travel from Norway to Svalbard you have to clear customs. It’s treated as a separate thing.

(http://www.autoadmit.com/thread.php?thread_id=5868609&forum_id=2/#49897972)