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Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnmlno0lem 21201 Lemma for nmlno0i 21202. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
CV       CV

Theoremnmlno0i 21202 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)

Theoremnmlno0 21203 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)

Theoremnmlnoubi 21204* An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremnmlnogt0 21205 The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)

Theoremlnon0 21206* The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)

Theoremnmblolbii 21207 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremnmblolbi 21208 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremisblo3i 21209* 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 21210* Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremblometi 21211 Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)

Theoremblocnilem 21212 Lemma for blocni 21213 and lnocni 21214. 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 21213 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 21214 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 21215 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 21216 A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)

Theoremajfval 21217* The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhmoval 21218* 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 21219 The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

15.6  Inner product (pre-Hilbert) spaces

15.6.1  Definition and basic properties

Syntaxccphlo 21220 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).

Definitiondf-ph 21221* 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 21222 Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphrel 21223 The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphnvi 21224 Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremisphg 21225* 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 21226 A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

15.6.2  Examples of pre-Hilbert spaces

Theoremcncph 21227 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 21228 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremelimphu 21229 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 6-May-2007.) (New usage is discouraged.)

15.6.3  Properties of pre-Hilbert spaces

Theoremisph 21230* The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
CV

Theoremphpar2 21231 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremphpar 21232 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremip0i 21233 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 21234 Lemma for ip1i 21235. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
CV

Theoremip1i 21235 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremip2i 21236 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdirilem 21237 Lemma for ipdiri 21238. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdiri 21238 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem1 21239 Lemma for ipassi 21249. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem2 21240 Lemma for ipassi 21249. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem3 21241 Lemma for ipassi 21249. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem4 21242 Lemma for ipassi 21249. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem5 21243 Lemma for ipassi 21249. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem7 21244* Lemma for ipassi 21249. Show that is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
fld

Theoremipasslem8 21245* Lemma for ipassi 21249. By ipasslem5 21243, is 0 for all ; since it is continuous and is dense in by qdensere2 18135, we conclude is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)

Theoremipasslem9 21246 Lemma for ipassi 21249. Conclude from ipasslem8 21245 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)

Theoremipasslem10 21247 Lemma for ipassi 21249. Show the inner product associative law for the imaginary number . (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
CV

Theoremipasslem11 21248 Lemma for ipassi 21249. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremipassi 21249 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdir 21250 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdi 21251 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremip2dii 21252 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)

Theoremdipass 21253 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipassr 21254 "Associative" law for second argument of inner product (compare dipass 21253). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)

Theoremdipassr2 21255 "Associative" law for inner product. Conjugate version of dipassr 21254. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdir 21256 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdi 21257 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theorempythi 21258 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 21259 Lemma for sii 21262. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiilem2 21260 Lemma for sii 21262. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiii 21261 Inference from sii 21262. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
CV

Theoremsii 21262 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 21529, bcsiALT 21588, bcsiHIL 21589, csbrn 25628. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
CV

Theoremsspph 21263 A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Theoremipblnfi 21264* 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 21265* 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 21266* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajmoi 21267* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfuni 21268 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfun 21269 The adjoint function is a function. This is not immediately apparent from df-aj 21158 but results from the uniqueness shown by ajmoi 21267. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Theoremajval 21270* Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

15.7  Complex Banach spaces

15.7.1  Definition and basic properties

Syntaxccbn 21271 Extend class notation with the class of all complex Banach spaces.

Definitiondf-cbn 21272 Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)

Theoremiscbn 21273 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 21274 The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theorembnnv 21275 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnrel 21276 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnsscmcl 21277 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.)

15.7.2  Examples of complex Banach spaces

Theoremcnbn 21278 The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.)

15.7.3  Uniform Boundedness Theorem

Theoremubthlem1 21279* Lemma for ubth 21282. 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 18584, 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 21280* Lemma for ubth 21282. 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 21281* Lemma for ubth 21282. Prove the reverse implication, using nmblolbi 21208. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
CV

Theoremubth 21282* 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

15.7.4  Minimizing Vector Theorem

Theoremminvecolem1 21283* Lemma for minveco 21293. 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 21284* Lemma for minveco 21293. 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 21285* Lemma for minveco 21293. 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 21286* Lemma for minveco 21293. is convergent in the subspace topology on . (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4b 21287* Lemma for minveco 21293. 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 21288* Lemma for minveco 21293. The infimum of the distances to is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4 21289* Lemma for minveco 21293. 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 21290* Lemma for minveco 21293. Discharge the assumption about the sequence by applying countable choice ax-cc 7945. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem6 21291* Lemma for minveco 21293. Any minimal point is less than away from . (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem7 21292* Lemma for minveco 21293. 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 21293* 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

15.8  Complex Hilbert spaces

15.8.1  Definition and basic properties

Syntaxchlo 21294 Extend class notation with the class of all complex Hilbert spaces.

Definitiondf-hlo 21295 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 21296 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 21297 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Theoremhlph 21298 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 21299 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremhlnv 21300 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

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