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Theorem List for Metamath Proof Explorer - 25401-25500   *Has distinct variable group(s)
TypeLabelDescription
Statement

19.4.3  Properties of pre-Hilbert spaces

Theoremisph 25401* The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
CV

Theoremphpar2 25402 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremphpar 25403 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremip0i 25404 A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where is either 1 or -1 to represent +-1. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
CV

Theoremip1ilem 25405 Lemma for ip1i 25406. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
CV

Theoremip1i 25406 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremip2i 25407 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdirilem 25408 Lemma for ipdiri 25409. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdiri 25409 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem1 25410 Lemma for ipassi 25420. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem2 25411 Lemma for ipassi 25420. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem3 25412 Lemma for ipassi 25420. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem4 25413 Lemma for ipassi 25420. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem5 25414 Lemma for ipassi 25420. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem7 25415* Lemma for ipassi 25420. Show that is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
fld

Theoremipasslem8 25416* Lemma for ipassi 25420. By ipasslem5 25414, is 0 for all ; since it is continuous and is dense in by qdensere2 21032, we conclude is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)

Theoremipasslem9 25417 Lemma for ipassi 25420. Conclude from ipasslem8 25416 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)

Theoremipasslem10 25418 Lemma for ipassi 25420. Show the inner product associative law for the imaginary number . (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
CV

Theoremipasslem11 25419 Lemma for ipassi 25420. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremipassi 25420 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdir 25421 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdi 25422 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremip2dii 25423 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)

Theoremdipass 25424 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipassr 25425 "Associative" law for second argument of inner product (compare dipass 25424). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)

Theoremdipassr2 25426 "Associative" law for inner product. Conjugate version of dipassr 25425. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdir 25427 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdi 25428 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theorempythi 25429 The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space . The square of the norm of the sum of two orthogonal vectors (i.e. whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. This is Metamath 100 proof #4. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
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Theoremsiilem1 25430 Lemma for sii 25433. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiilem2 25431 Lemma for sii 25433. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiii 25432 Inference from sii 25433. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
CV

Theoremsii 25433 Schwarz inequality. Part of Lemma 3-2.1(a) of [Kreyszig] p. 137. This is also called the Cauchy-Schwarz inequality by some authors and Bunjakovaskij-Cauchy-Schwarz inequality by others. See also theorems bcseqi 25701, bcsiALT 25760, bcsiHIL 25761, csbren 21556. This is Metamath 100 proof #78. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
CV

Theoremsspph 25434 A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Theoremipblnfi 25435* A function generated by varying the first argument of an inner product (with its second argument a fixed vector ) is a bounded linear functional, i.e. a bounded linear operator from the vector space to . (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremip2eqi 25436* Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremphoeqi 25437* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajmoi 25438* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfuni 25439 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfun 25440 The adjoint function is a function. This is not immediately apparent from df-aj 25329 but results from the uniqueness shown by ajmoi 25438. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Theoremajval 25441* Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

19.5  Complex Banach spaces

19.5.1  Definition and basic properties

Syntaxccbn 25442 Extend class notation with the class of all complex Banach spaces.

Definitiondf-cbn 25443 Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)

Theoremiscbn 25444 A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)

Theoremcbncms 25445 The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theorembnnv 25446 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnrel 25447 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnsscmcl 25448 A subspace of a Banach space is a Banach space iff it is closed in the norm-induced metric of the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

19.5.2  Examples of complex Banach spaces

Theoremcnbn 25449 The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.)

19.5.3  Uniform Boundedness Theorem

Theoremubthlem1 25450* Lemma for ubth 25453. The function exhibits a countable collection of sets that are closed, being the inverse image under of the closed ball of radius , and by assumption they cover . Thus, by the Baire Category theorem bcth2 21499, for some the set has an interior, meaning that there is a closed ball in the set. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
CV

Theoremubthlem2 25451* Lemma for ubth 25453. Given that there is a closed ball in , for any , we have and , so both of these have and so , which is our desired uniform bound. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
CV

Theoremubthlem3 25452* Lemma for ubth 25453. Prove the reverse implication, using nmblolbi 25379. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
CV

Theoremubth 25453* Uniform Boundedness Theorem, also called the Banach-Steinhaus Theorem. Let be a collection of bounded linear operators on a Banach space. If, for every vector , the norms of the operators' values are bounded, then the operators' norms are also bounded. Theorem 4.7-3 of [Kreyszig] p. 249. See also http://en.wikipedia.org/wiki/Uniform_boundedness_principle. (Contributed by NM, 7-Nov-2007.) (Proof shortened by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
CV

19.5.4  Minimizing Vector Theorem

Theoremminvecolem1 25454* Lemma for minveco 25464. The set of all distances from points of to are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem2 25455* Lemma for minveco 25464. Any two points and in are close to each other if they are close to the infimum of distance to . (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem3 25456* Lemma for minveco 25464. The sequence formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4a 25457* Lemma for minveco 25464. is convergent in the subspace topology on . (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4b 25458* Lemma for minveco 25464. The convergent point of the cauchy sequence is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4c 25459* Lemma for minveco 25464. The infimum of the distances to is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4 25460* Lemma for minveco 25464. The convergent point of the cauchy sequence attains the minimum distance, and so is closer to than any other point in . (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem5 25461* Lemma for minveco 25464. Discharge the assumption about the sequence by applying countable choice ax-cc 8806. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem6 25462* Lemma for minveco 25464. Any minimal point is less than away from . (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem7 25463* Lemma for minveco 25464. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminveco 25464* Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace that minimizes the distance to an arbitrary vector in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

19.6  Complex Hilbert spaces

19.6.1  Definition and basic properties

Syntaxchlo 25465 Extend class notation with the class of all complex Hilbert spaces.

Definitiondf-hlo 25466 Define the class of all complex Hilbert spaces. A Hilbert space is a Banach space which is also an inner product space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Theoremishlo 25467 The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Theoremhlobn 25468 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Theoremhlph 25469 Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (New usage is discouraged.)

Theoremhlrel 25470 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremhlnv 25471 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremhlnvi 25472 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Theoremhlvc 25473 Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlcmet 25474 The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlmet 25475 The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlpar2 25476 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
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Theoremhlpar 25477 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
CV

19.6.2  Standard axioms for a complex Hilbert space

Theoremhlex 25478 The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhladdf 25479 Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlcom 25480 Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlass 25481 Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhl0cl 25482 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhladdid 25483 Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulf 25484 Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulid 25485 Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulass 25486 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhldi 25487 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhldir 25488 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmul0 25489 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlipf 25490 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)

Theoremhlipcj 25491 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipdir 25492 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipass 25493 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipgt0 25494 The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlcompl 25495 Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)

19.6.3  Examples of complex Hilbert spaces

Theoremcnchl 25496 The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

19.6.4  Subspaces

Theoremssphl 25497 A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.)

19.6.5  Hellinger-Toeplitz Theorem

Theoremhtthlem 25498* Lemma for htth 25499. The collection , which consists of functions for each in the unit ball, is a collection of bounded linear functions by ipblnfi 25435, so by the Uniform Boundedness theorem ubth 25453, there is a uniform bound on for all in the unit ball. Then , so and is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
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Theoremhtth 25499* Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)

This part contains the definitions and theorems used by the Hilbert Space Explorer (HSE), http://us.metamath.org/mpeuni/mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page.

Future development should instead work with general Hilbert spaces as defined by df-hil 18497; note that df-hil 18497 uses extensible structures.

The intent is for this deprecated section to be deleted once all its theorems have been translated into extensible structure versions (or are not useful). Many of the theorems in this section have already been translated to extensible structure versions, but there is still a lot in this section that might be useful for future reference. It is much easier to translate these by hand from this section than to start from scratch from textbook proofs, since the HSE omits no details.

20.1  Axiomatization of complex pre-Hilbert spaces

20.1.1  Basic Hilbert space definitions

Syntaxchil 25500 Extend class notation with Hilbert vector space.

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