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Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremphpar2 21401 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremphpar 21402 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremip0i 21403 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 21404 Lemma for ip1i 21405. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
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Theoremip1i 21405 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremip2i 21406 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdirilem 21407 Lemma for ipdiri 21408. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdiri 21408 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem1 21409 Lemma for ipassi 21419. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem2 21410 Lemma for ipassi 21419. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem3 21411 Lemma for ipassi 21419. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem4 21412 Lemma for ipassi 21419. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem5 21413 Lemma for ipassi 21419. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem7 21414* Lemma for ipassi 21419. Show that is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
fld

Theoremipasslem8 21415* Lemma for ipassi 21419. By ipasslem5 21413, is 0 for all ; since it is continuous and is dense in by qdensere2 18303, we conclude is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)

Theoremipasslem9 21416 Lemma for ipassi 21419. Conclude from ipasslem8 21415 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)

Theoremipasslem10 21417 Lemma for ipassi 21419. Show the inner product associative law for the imaginary number . (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
CV

Theoremipasslem11 21418 Lemma for ipassi 21419. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremipassi 21419 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdir 21420 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdi 21421 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremip2dii 21422 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)

Theoremdipass 21423 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipassr 21424 "Associative" law for second argument of inner product (compare dipass 21423). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)

Theoremdipassr2 21425 "Associative" law for inner product. Conjugate version of dipassr 21424. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdir 21426 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdi 21427 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theorempythi 21428 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. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
CV

Theoremsiilem1 21429 Lemma for sii 21432. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiilem2 21430 Lemma for sii 21432. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiii 21431 Inference from sii 21432. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
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Theoremsii 21432 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 21699, bcsiALT 21758, bcsiHIL 21759, csbrn 26462. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
CV

Theoremsspph 21433 A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Theoremipblnfi 21434* 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 21435* 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 21436* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajmoi 21437* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfuni 21438 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfun 21439 The adjoint function is a function. This is not immediately apparent from df-aj 21328 but results from the uniqueness shown by ajmoi 21437. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Theoremajval 21440* Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

16.5  Complex Banach spaces

16.5.1  Definition and basic properties

Syntaxccbn 21441 Extend class notation with the class of all complex Banach spaces.

Definitiondf-cbn 21442 Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)

Theoremiscbn 21443 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 21444 The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theorembnnv 21445 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnrel 21446 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnsscmcl 21447 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.)

16.5.2  Examples of complex Banach spaces

Theoremcnbn 21448 The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.)

16.5.3  Uniform Boundedness Theorem

Theoremubthlem1 21449* Lemma for ubth 21452. 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 18752, 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 21450* Lemma for ubth 21452. 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 21451* Lemma for ubth 21452. Prove the reverse implication, using nmblolbi 21378. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
CV

Theoremubth 21452* 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

16.5.4  Minimizing Vector Theorem

Theoremminvecolem1 21453* Lemma for minveco 21463. 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 21454* Lemma for minveco 21463. 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 21455* Lemma for minveco 21463. 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 21456* Lemma for minveco 21463. is convergent in the subspace topology on . (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4b 21457* Lemma for minveco 21463. 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 21458* Lemma for minveco 21463. The infimum of the distances to is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4 21459* Lemma for minveco 21463. 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 21460* Lemma for minveco 21463. Discharge the assumption about the sequence by applying countable choice ax-cc 8061. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem6 21461* Lemma for minveco 21463. Any minimal point is less than away from . (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem7 21462* Lemma for minveco 21463. 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 21463* 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

16.6  Complex Hilbert spaces

16.6.1  Definition and basic properties

Syntaxchlo 21464 Extend class notation with the class of all complex Hilbert spaces.

Definitiondf-hlo 21465 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 21466 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 21467 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Theoremhlph 21468 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 21469 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremhlnv 21470 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremhlnvi 21471 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Theoremhlvc 21472 Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlcmet 21473 The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlmet 21474 The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlpar2 21475 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
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Theoremhlpar 21476 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
CV

16.6.2  Standard axioms for a complex Hilbert space

Theoremhlex 21477 The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhladdf 21478 Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlcom 21479 Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlass 21480 Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhl0cl 21481 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhladdid 21482 Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulf 21483 Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulid 21484 Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulass 21485 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhldi 21486 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhldir 21487 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmul0 21488 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlipf 21489 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)

Theoremhlipcj 21490 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipdir 21491 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipass 21492 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipgt0 21493 The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlcompl 21494 Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)

16.6.3  Examples of complex Hilbert spaces

Theoremcnchl 21495 The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

16.6.4  Subspaces

Theoremssphl 21496 A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.)

16.6.5  Hellinger-Toeplitz Theorem

Theoremhtthlem 21497* Lemma for htth 21498. The collection , which consists of functions for each in the unit ball, is a collection of bounded linear functions by ipblnfi 21434, so by the Uniform Boundedness theorem ubth 21452, 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.)
CV

Theoremhtth 21498* 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 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)

This part contains the definitions and theorems used by the Hilbert Space Explorer 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 work with general Hilbert spaces as defined by df-hil 16604.

17.1  Axiomatization of complex pre-Hilbert spaces

17.1.1  Basic Hilbert space definitions

Syntaxchil 21499 Extend class notation with Hilbert vector space.

Syntaxcva 21500 Extend class notation with vector addition in Hilbert space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition caddc 8740.

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