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==Root lineal==
Let <math>\mathbf{X}</math> be a [[Banach space]]. We recall that the [[Generalized_eigenvector#Root_lineal_of_a_linear_operator_in_a_Banach_space|root lineal]] <math>\mathfrak{L}_\lambda(A)</math> of a linear operator <math>A:\,\mathbf{X}\to\mathbf{X}</math> with ___domain <math>\mathfrak{D}(A)</math> corresponding to the eigenvalue <math>\lambda\in\sigma_p(A)</math> is defined as
:<math>\mathfrak{L}_\lambda(A)=\cup_{k\in\N}\{x\in\mathfrak{D}(A):\,(A-\lambda I_{\mathbf{X}})^j x\in\mathfrak{D}(A)\,\forall j\in\N,\,j\le k;\, (A-\lambda I_{\mathbf{X}})^k x=0\}\subset\mathbf{X}</math>.
This set is a [[linear manifold]] but not necessarily a [[vector space]], since it is not necessarily closed in <math>\mathbf{X}</math>. If this set is closed (for example, when it is finite-dimensional), it is called the [[generalized eigenspace]] of <math>A</math> corresponding to the eigenvalue <math>\lambda</math>.
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==Definition==
An [[eigenvalue]] <math>\lambda\in\sigma_p(A)</math> of a [[Unbounded_operator#Closed_linear_operators|closed linear operator]] <math>A:\,\mathbf{X}\to\mathbf{X}</math> in the [[Banach space]] <math>\mathbf{X}</math> with [[Unbounded_operator#Definitions_and_basic_properties|___domain]] <math>\mathfrak{D}(A)\subset\mathbf{X}</math> is called ''normal'' if the following two conditions are satisfied:
# The [[algebraic multiplicity]] of <math>\lambda</math> is finite: <math>\nu=\dim\mathfrak{L}_\lambda(A)<\infty</math>, where <math>\mathfrak{L}_\lambda(A)</math> is the [[Generalized_eigenvector#Root_lineal_of_a_linear_operator_in_a_Banach_space|root lineal]] of <math>A</math> corresponding to the eigenvalue <math>\lambda</math>;
# The space <math>\mathbf{X}</math> could be decomposed into a direct sum <math>\mathbf{X}=\mathfrak{L}_\lambda(A)\oplus \mathfrak{N}_\lambda</math>, where <math>\mathfrak{N}_\lambda</math> is an [[invariant subspace]] of <math>A</math> in which <math>A-\lambda I_{\mathbf{X}}</math> has a bounded inverse.
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