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

Theoremlnon0 22301* The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)

Theoremnmblolbii 22302 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremnmblolbi 22303 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremisblo3i 22304* The predicate "is a bounded linear operator." Definition 2.7-1 of [Kreyszig] p. 91. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremblo3i 22305* Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremblometi 22306 Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)

Theoremblocnilem 22307 Lemma for blocni 22308 and lnocni 22309. If a linear operator is continuous at any point, it is bounded. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)

Theoremblocni 22308 A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)

Theoremlnocni 22309 If a linear operator is continuous at any point, it is continuous everywhere. Theorem 2.7-9(b) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (New usage is discouraged.)

Theoremblocn 22310 A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)

Theoremblocn2 22311 A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)

Theoremajfval 22312* The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhmoval 22313* The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremishmo 22314 The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

17.4  Inner product (pre-Hilbert) spaces

17.4.1  Definition and basic properties

Syntaxccphlo 22315 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).

Definitiondf-ph 22316* Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is , the scalar product is , and the norm is . An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphnv 22317 Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphrel 22318 The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphnvi 22319 Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremisphg 22320* The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is , the scalar product is , and the norm is . An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphop 22321 A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

17.4.2  Examples of pre-Hilbert spaces

Theoremcncph 22322 The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 7-Nov-2013.) (New usage is discouraged.)

Theoremelimph 22323 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremelimphu 22324 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 6-May-2007.) (New usage is discouraged.)

17.4.3  Properties of pre-Hilbert spaces

Theoremisph 22325* The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
CV

Theoremphpar2 22326 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremphpar 22327 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremip0i 22328 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 22329 Lemma for ip1i 22330. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
CV

Theoremip1i 22330 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremip2i 22331 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdirilem 22332 Lemma for ipdiri 22333. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdiri 22333 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem1 22334 Lemma for ipassi 22344. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem2 22335 Lemma for ipassi 22344. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem3 22336 Lemma for ipassi 22344. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem4 22337 Lemma for ipassi 22344. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem5 22338 Lemma for ipassi 22344. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem7 22339* Lemma for ipassi 22344. Show that is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
fld

Theoremipasslem8 22340* Lemma for ipassi 22344. By ipasslem5 22338, is 0 for all ; since it is continuous and is dense in by qdensere2 18830, we conclude is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)

Theoremipasslem9 22341 Lemma for ipassi 22344. Conclude from ipasslem8 22340 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)

Theoremipasslem10 22342 Lemma for ipassi 22344. Show the inner product associative law for the imaginary number . (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
CV

Theoremipasslem11 22343 Lemma for ipassi 22344. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremipassi 22344 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdir 22345 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdi 22346 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremip2dii 22347 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)

Theoremdipass 22348 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipassr 22349 "Associative" law for second argument of inner product (compare dipass 22348). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)

Theoremdipassr2 22350 "Associative" law for inner product. Conjugate version of dipassr 22349. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdir 22351 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdi 22352 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theorempythi 22353 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 22354 Lemma for sii 22357. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiilem2 22355 Lemma for sii 22357. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiii 22356 Inference from sii 22357. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
CV

Theoremsii 22357 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 22624, bcsiALT 22683, bcsiHIL 22684, csbrn 26458. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
CV

Theoremsspph 22358 A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Theoremipblnfi 22359* 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 22360* 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 22361* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajmoi 22362* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfuni 22363 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfun 22364 The adjoint function is a function. This is not immediately apparent from df-aj 22253 but results from the uniqueness shown by ajmoi 22362. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Theoremajval 22365* 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 22366 Extend class notation with the class of all complex Banach spaces.

Definitiondf-cbn 22367 Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)

Theoremiscbn 22368 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 22369 The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theorembnnv 22370 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnrel 22371 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnsscmcl 22372 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 22373 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 22374* Lemma for ubth 22377. 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 19285, 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 22375* Lemma for ubth 22377. 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 22376* Lemma for ubth 22377. Prove the reverse implication, using nmblolbi 22303. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
CV

Theoremubth 22377* 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

17.5.4  Minimizing Vector Theorem

Theoremminvecolem1 22378* Lemma for minveco 22388. 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 22379* Lemma for minveco 22388. 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 22380* Lemma for minveco 22388. 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 22381* Lemma for minveco 22388. is convergent in the subspace topology on . (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4b 22382* Lemma for minveco 22388. 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 22383* Lemma for minveco 22388. The infimum of the distances to is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4 22384* Lemma for minveco 22388. 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 22385* Lemma for minveco 22388. Discharge the assumption about the sequence by applying countable choice ax-cc 8317. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem6 22386* Lemma for minveco 22388. Any minimal point is less than away from . (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem7 22387* Lemma for minveco 22388. 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 22388* 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

17.6  Complex Hilbert spaces

17.6.1  Definition and basic properties

Syntaxchlo 22389 Extend class notation with the class of all complex Hilbert spaces.

Definitiondf-hlo 22390 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 22391 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 22392 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Theoremhlph 22393 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 22394 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremhlnv 22395 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremhlnvi 22396 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Theoremhlvc 22397 Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlcmet 22398 The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlmet 22399 The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlpar2 22400 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
CV

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