Invariant subspace

An invariant subspace is a subspace that each transformation in a specified family maps into itself. For a linear operator, this means applying the operator to any vector in the subspace produces another vector in it.

Linear operator: A map between vector spaces that preserves vector addition and scalar multiplication. An invariant subspace is defined by how a linear operator maps its vectors.
Eigenvector: A nonzero vector that a linear operator maps to a scalar multiple of itself. Its one-dimensional span is an invariant subspace.
Vector space: A set with addition and scalar multiplication satisfying the vector-space axioms. A subspace is defined inside a vector space.
Invariant subspace problem: The question of whether every bounded linear operator on a complex infinite-dimensional Banach space has a nontrivial closed invariant subspace. This famous problem asks whether invariant subspaces must exist in a broad setting.
Eigenspace: The subspace of vectors associated with a fixed eigenvalue, together with the zero vector. Unlike a general invariant subspace, an eigenspace consists of vectors scaled by one eigenvalue.
Restriction of a linear operator: The map obtained by limiting a function's domain to a specified subset. An operator on an invariant subspace is its restriction to that subspace.
Jordan normal form: A canonical matrix form over an algebraically closed field, built from Jordan blocks. Chains of invariant subspaces underlie the block decomposition.
Subspace: A subset of a vector space that is itself a vector space under the inherited operations. Invariance asks whether transformations preserve this kind of subset.
Enflo's invariant subspace theorem: Per Enflo's construction of a Banach space and bounded operator with no nontrivial closed invariant subspace. It disproved the universal existence claim for arbitrary Banach spaces.
Cyclic subspace: The span of all images of one vector under an operator and its powers. It is generated by an orbit and is invariant, rather than specified in advance.

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