| Riesz Representation Theorem |
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THE HILBERT SPACE REPRESENTATION THEOREM This theorem establishes an important connection between a , the two are Isometrically isomorphic; if the ground field is the Complex Numbers , the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next. Let ''H'' be a Hilbert space, and let H ' denote its dual space, consisting of all Continuous Linear Operator s from ''H'' into the base field R or '''C'''. If ''x'' is an element of ''H'', then the function φ''x'' defined by : where ( , ) denotes the Inner Product of the Hilbert space, is an element of H '. The Riesz representation theorem states that every element of H ' can be written uniquely in this form: Theorem. The mapping : is an isometric (anti-) isomorphism, meaning that:
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The Inverse Map Of &Phi Can Be Described As Follows Given An Element &phi Of <i>H</i> ', The Orthogonal Complement Of The Kernel Of &phi Is A One-dimensional Subspace Of ''H'' Take A Non-zero Element ''z'' In That Subspace, And Set ''x'' |
&phi(''z'') / ''z''<sup>2</sup> ยท ''z'' Then &Phi(''x'') = &phi |
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In The Mathematical Treatment Of
| "http://wwwinformationdelightinfo/encyclopedia/entry/quantum_mechanics" class="copylinks">Quantum Mechanics , the theorem can be seen as a justification for the popular Bra-ket Notation When the theorem holds, every ket <math>\psi
angle</math> has a corresponding bra <math>\langle\psi</math>, and the correspondence is unambiguous However, there are Topological Vector Space s, such as Nuclear Space s, where the Riesz repesentation theorem does not hold, in which case the bra-ket notation can become awkward |
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:<math> \\psi\ |
\mu(X)</math> |
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