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Theorem List for Metamath Proof Explorer - 22301-22400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdipassr2 22301 "Associative" law for inner product. Conjugate version of dipassr 22300. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdir 22302 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdi 22303 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theorempythi 22304 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.)
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Theoremsiilem1 22305 Lemma for sii 22308. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
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Theoremsiilem2 22306 Lemma for sii 22308. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
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Theoremsiii 22307 Inference from sii 22308. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
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Theoremsii 22308 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 22575, bcsiALT 22634, bcsiHIL 22635, csbrn 26346. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
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Theoremsspph 22309 A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Theoremipblnfi 22310* 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 22311* 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 22312* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajmoi 22313* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfuni 22314 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfun 22315 The adjoint function is a function. This is not immediately apparent from df-aj 22204 but results from the uniqueness shown by ajmoi 22313. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Theoremajval 22316* Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

17.5  Complex Banach spaces

17.5.1  Definition and basic properties

Syntaxccbn 22317 Extend class notation with the class of all complex Banach spaces.

Definitiondf-cbn 22318 Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)

Theoremiscbn 22319 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 22320 The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theorembnnv 22321 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnrel 22322 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnsscmcl 22323 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.)

17.5.2  Examples of complex Banach spaces

Theoremcnbn 22324 The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.)

17.5.3  Uniform Boundedness Theorem

Theoremubthlem1 22325* Lemma for ubth 22328. 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 19236, 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.)
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Theoremubthlem2 22326* Lemma for ubth 22328. 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.)
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Theoremubthlem3 22327* Lemma for ubth 22328. Prove the reverse implication, using nmblolbi 22254. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
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Theoremubth 22328* 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.)
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17.5.4  Minimizing Vector Theorem

Theoremminvecolem1 22329* Lemma for minveco 22339. 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.)
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Theoremminvecolem2 22330* Lemma for minveco 22339. 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.)
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Theoremminvecolem3 22331* Lemma for minveco 22339. The sequence formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (New usage is discouraged.)
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Theoremminvecolem4a 22332* Lemma for minveco 22339. is convergent in the subspace topology on . (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
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Theoremminvecolem4b 22333* Lemma for minveco 22339. 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.)
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Theoremminvecolem4c 22334* Lemma for minveco 22339. The infimum of the distances to is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
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Theoremminvecolem4 22335* Lemma for minveco 22339. 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.)
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Theoremminvecolem5 22336* Lemma for minveco 22339. Discharge the assumption about the sequence by applying countable choice ax-cc 8271. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
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Theoremminvecolem6 22337* Lemma for minveco 22339. Any minimal point is less than away from . (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
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Theoremminvecolem7 22338* Lemma for minveco 22339. 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.)
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Theoremminveco 22339* 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.)
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17.6  Complex Hilbert spaces

17.6.1  Definition and basic properties

Syntaxchlo 22340 Extend class notation with the class of all complex Hilbert spaces.

Definitiondf-hlo 22341 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 22342 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 22343 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Theoremhlph 22344 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 22345 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremhlnv 22346 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremhlnvi 22347 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Theoremhlvc 22348 Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlcmet 22349 The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlmet 22350 The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlpar2 22351 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
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Theoremhlpar 22352 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
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17.6.2  Standard axioms for a complex Hilbert space

Theoremhlex 22353 The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhladdf 22354 Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlcom 22355 Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlass 22356 Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhl0cl 22357 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhladdid 22358 Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulf 22359 Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulid 22360 Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulass 22361 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhldi 22362 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhldir 22363 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmul0 22364 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlipf 22365 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)

Theoremhlipcj 22366 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipdir 22367 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipass 22368 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipgt0 22369 The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlcompl 22370 Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)

17.6.3  Examples of complex Hilbert spaces

Theoremcnchl 22371 The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

17.6.4  Subspaces

Theoremssphl 22372 A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.)

17.6.5  Hellinger-Toeplitz Theorem

Theoremhtthlem 22373* Lemma for htth 22374. The collection , which consists of functions for each in the unit ball, is a collection of bounded linear functions by ipblnfi 22310, so by the Uniform Boundedness theorem ubth 22328, 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 22374* 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 18  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 16886.

18.1  Axiomatization of complex pre-Hilbert spaces

18.1.1  Basic Hilbert space definitions

Syntaxchil 22375 Extend class notation with Hilbert vector space.

Syntaxcva 22376 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 8949.

Syntaxcsm 22377 Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.

Syntaxcsp 22378 Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of and is usually written but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 3783.

Syntaxcno 22379 Extend class notation with the norm function in Hilbert space. In the literature, the norm of is usually written "|| ||", but we use function notation to take advantage of our existing theorems about functions.

Syntaxc0v 22380 Extend class notation with zero vector in Hilbert space.

Syntaxcmv 22381 Extend class notation with vector subtraction in Hilbert space.

Syntaxccau 22382 Extend class notation with set of Cauchy sequences in Hilbert space.

Syntaxchli 22383 Extend class notation with convergence relation in Hilbert space.

Syntaxcsh 22384 Extend class notation with set of subspaces of a Hilbert space.

Syntaxcch 22385 Extend class notation with set of closed subspaces of a Hilbert space.

Syntaxcort 22386 Extend class notation with orthogonal complement in .

Syntaxcph 22387 Extend class notation with subspace sum in .

Syntaxcspn 22388 Extend class notation with subspace span in .

Syntaxchj 22389 Extend class notation with join in .

Syntaxchsup 22390 Extend class notation with supremum of a collection in .

Syntaxc0h 22391 Extend class notation with zero of .

Syntaxccm 22392 Extend class notation with the commutes relation on a Hilbert lattice.

Syntaxcpjh 22393 Extend class notation with set of projections on a Hilbert space.

Syntaxchos 22394 Extend class notation with sum of Hilbert space operators.

Syntaxchot 22395 Extend class notation with scalar product of a Hilbert space operator.

Syntaxchod 22396 Extend class notation with difference of Hilbert space operators.

Syntaxchfs 22397 Extend class notation with sum of Hilbert space functionals.

Syntaxchft 22398 Extend class notation with scalar product of Hilbert space functional.

Syntaxch0o 22399 Extend class notation with the Hilbert space zero operator.

Syntaxchio 22400 Extend class notation with Hilbert space identity operator.

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