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The David Hilbert , generalizes the notion of Euclidean Space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces. In more formal terms, a Hilbert space is an Inner Product Space — an abstract Vector Space in which distances and angles can be measured — which is " Complete ", meaning that if a sequence of vectors approaches a Limit , then that limit is guaranteed to be in the space as well. Hilbert spaces arise naturally and frequently in Mathematics , Physics , and Engineering , typically as infinite-dimensional Function Space s. They are indispensable tools in the theories of Partial Differential Equation s, Quantum Mechanics , and Signal Processing . The recognition of a common algebraic structure within these diverse fields generated a greater conceptual understanding, and the success of Hilbert space methods ushered in a very fruitful era for Functional Analysis . Geometric intuition plays an important role in many aspects of Hilbert space theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an Orthonormal Basis , in analogy with cartesian coordinates in the plane. This means that Hilbert space can also usefully be thought of in terms of Infinite Sequence s that are Square-summable . Linear Operator s on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions. MOTIVATION AND INTUITIVE MEANING Ordinary Euclidean Space R3 serves as a model for the more abstract notion of a Hilbert space. In the Euclidean space, the Distance between points and the Angle between vectors can be expressed via the Dot Product , a certain Bilinear Operation on vectors with values in Real Numbers . Many problems from Analytic Geometry can be reworded and solved using the dot product, for example, "When are two lines Orthogonal ?" or "Which point on a given plane is closest to the origin?" In a Hilbert space, the fundamental objects are abstractions of vectors, whose nature is unimportant (they may be, for example, sequences or functions of some kind). Those abstract vectors can be added and multiplied by a scalar, and an analogue of the dot product is defined for them. The algebraic operations on vectors in a Hilbert space have familiar properties, like Commutativity and Distributivity . In addition, the technical requirement of completeness ensures that certain Limit s exist. This last property is always true for finite-dimensional Inner Product Space s, but needs to be stated as an additional assumption in the more general case. While the definition of a Hilbert space given below may appear complicated, due to a large number of consistency axioms, the basic intuition behind Hilbert spaces is amazingly simple: : ''In a large range of physical and mathematical situations, a linear problem can be stated within a certain Hilbert space and analyzed in simple geometrical terms''. In particular, this principle applies to solving linear can be decomposed into infinitely many independent parts, which is closely analogous to the way of representing a vector from R3 as a Linear Combination of three orthogonal vectors. Similar considerations apply to other equations of mathematical physics, notably, the Wave Equation and Helmholtz Equation . The success of the theory of Hilbert spaces is due in part to the striking fact that : ''although they may differ in origin and appearance, most Hilbert spaces considered in physics and mathematics are just multiple manifestations of a single Separable Hilbert space''. One way to comprehend this proceeds by introducing a system of coordinates into a given Hilbert space using the notion of Orthonormal Basis described below. As a consequence of the uniqueness principle, a theorem stated in abstract terms and valid in one of these spaces will hold in all of them. DEFINITION A Real or Complex Hilbert space is a real or complex Inner Product Space that is a '''complete''' normed space ( Banach Space ) under the norm defined by the inner product. Remarks | ||
| #:<math>\x\ | \sqrt{\langle x,x
angle} </math> |
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| #: A | "http://wwwinformationdelightinfo/information/entry/sequence" class="copylinks">Sequence {''v''<sub>''n''</sub>} is a Cauchy Sequence if for every positive real number ε there is a natural number ''N'' such that for all ''m'', ''n'' > ''N'', ''v''<sub>''n''</sub> – ''v''<sub>''m''</sub> < ε The space ''H'' is ''' Complete ''' with respect to this norm if every Cauchy sequence Converges to an element in the space |
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| #: <math>\\mathbf{u}+\mathbf{v}\^2+\\mathbf{u}-\mathbf{v}\^2 | 2(\\mathbf{u}\^2+\\mathbf{v}\^2)</math> |
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| :<math> \ell^2(B) | \big\{ x : B \xrightarrow{x} \mathbb{C} ext{ and } \sum_{b \in B} \leftx \left(b
ight)
ight^2 < \infty \big\}</math> |
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| # '''Normalization''': Every Element Of The Family Has Norm 1: ''e''<sub>''k''</sub> | 1 for all ''k'' in ''B'' |
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| '''Theorem''' The Orthogonal Projection P<sub>''V''</sub> Is A Self-adjoint Linear Operator On ''H'' Of Norm ≤ 1 With The Property P<sub>''V''</sub><sup>2</sup> | P<sub>''V''</sub> Moreover, any self-adjoint linear operator ''E'' such that ''E''<sup>2</sup> = ''E'' is of the form P<sub>''V''</sub>, where ''V'' is the range of ''E'' For every ''x'' in ''H'', P<sub>''V''</sub>(''x'') is the unique element ''v'' of ''V'' which minimizes the distance ''x'' - ''v'' |
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