The US Supreme Court recently rendered an important decision on the right to bear arms under the US constitution. The decision applies to government action, not private relationships such as landlord and tenant.
The Michigan Court of Appeals has ruled that a public housing landlord may enforce a lease provision prohibiting tenants from possessing firearms by evicting tenants that violate it. The court cited legal precedents that establish the right to bear arms is not absolute and is subject to reasonable regulation. 
It is important to note that although the tenant in question had apparent mental health issues, the court upheld the eviction of the tenant not as a nuisance or threat to other tenants, but upheld the landlord’s right to enforce the lease provision banning all firearms in the leased premises.
The Washington state Supreme Court held in a 2006 decision that a city government acting in a proprietary capacity as property owner could place conditions on firearms on property. The plaintiffs sought a temporary use permit from the city of Sequim for the use of city property for the purposes of a gun show. The city placed several conditions on the use of the city facilities for the show.
The Court noted that by issuing the permit “the city was leasing its property to PNSPA and acting in its private capacity as a property owner” and “that a municipal property owner like a private property owner may impose conditions related to firearms for the use of its property in order to protect its property interests”.
While the case is not a landlord-tenant case per se and there is little legal authority on point, the implication is that a private property owner may prohibit firearms.
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