数学领域泛函分析中，最著名的悬而未决的问题之一就是不变子空间问题，有时被乐观地称为不变子空间猜想。这个问题就是如下命题是否成立：

该命题对于所有2维以上有限维复矢量空间是成立的：一个线性算子（矩阵）的特征值是其特征多项式的零点；根据代数基本定理，这个多项式存在零点；一个对应的特征矢量可以张成一个不变子空间。该命题也很容易成立如果W不必是闭的：取任意H中非零矢量x并考虑H的由{T(x) : n ≥ 0}线性张成的子空间W.

虽然该猜想的一般情况未获证明，但已经可以列出命题成立的一些特殊情况：

In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. The original form of the problem as posed by Paul Halmos was in the special case of polynomials with compact square. This was resolved affirmatively, for a more general class of polynomially compact operators, by Allen R. Bernstein and Abraham Robinson in 1966 (see Non-standard analysis#Invariant subspace problem for a summary of the proof).