Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. It's essentially just an infinitely dimensional vector space. ﻿A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.
Spectrum (functional analysis)
For example, consider the Hilbert space formula_24, that consists of all bi-infinite sequences of real numbers
that have a finite sum of squares formula_26. The bilateral shift operator formula_2 simply displaces every element of the sequence by one position; namely if formula_28 then formula_29 for every integer formula_30. The eigenvalue equation formula_31 has no solution in this space, since it implies that all the values formula_32 have the same absolute value (if formula_33) or are a geometric progression (if formula_34); either way, the sum of their squares would not be finite. However, the operator formula_20 is not invertible if formula_36. For example, the sequence formula_37 such that formula_38 is in formula_24; but there is no sequence formula_40 in formula_24 such that formula_42 (that is, formula_43 for all formula_30).