Definition:Eigenvector

Definition
Let $H$ be a Hilbert space over $\Bbb F \in \left\{{\R, \C}\right\}$.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

Let $\alpha \in \Bbb F$ be an eigenvalue of $A$.

A nonzero vector $h \in H$ is said to be an eigenvector for $\alpha$ iff:


 * $h \in \operatorname{ker} \left({A - \alpha I}\right)$

That is, iff $Ah = \alpha h$.

Eigenspace
The eigenspace for an eigenvalue $\alpha$ is the set $\operatorname{ker} \left({A - \alpha I}\right)$.

By Kernel of Linear Transformation is Closed Linear Subspace, it is a closed linear subspace of $H$.