Invariant subspace problem

The invariant subspace problem is a question in mathematics about whether, for every linear operator (a kind of mathematical function) on a complex or real vector space, there always exists a nontrivial subspace that the operator keeps unchanged. In this context, a subspace is a subset that is closed under addition and scalar multiplication. An invariant subspace remains stable under the operator’s action. While this is true for many cases, it remains an open problem whether such invariant subspaces always exist for all operators, especially in infinite-dimensional spaces.