Unlike the simple case of a matrix, in infinite dimensional vector spaces, the spectrum may be continuous.
The quintessential example of that is the spectrum of the position operator in quantum mechanics, in which any real number is a possible eigenvalue, since the particle may be found in any position. The associated eigenvectors are the corresponding Dirac delta functions.
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In functional analysis, the notion of the spectrum of an operator is a fundamental concept that extends the idea of eigenvalues from finite-dimensional linear algebra to more general settings, particularly in the study of bounded linear operators on Banach spaces and Hilbert spaces.