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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremnmobndseqiOLD 22201* A bounded sequence determines a bounded operator. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
CV       CV

Theorembloval 22202* The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremisblo 22203 The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)

Theoremisblo2 22204 The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theorembloln 22205 A bounded operator is a linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremblof 22206 A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnmblore 22207 The norm of a bounded operator is a real number. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theorem0ofval 22208 The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theorem0oval 22209 Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)

Theorem0oo 22210 The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)

Theorem0lno 22211 The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremnmoo0 22212 The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)

Theorem0blo 22213 The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnmlno0lem 22214 Lemma for nmlno0i 22215. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
CV       CV

Theoremnmlno0i 22215 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)

Theoremnmlno0 22216 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)

Theoremnmlnoubi 22217* 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 22218 The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)

Theoremlnon0 22219* The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)

Theoremnmblolbii 22220 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremnmblolbi 22221 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremisblo3i 22222* 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 22223* Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremblometi 22224 Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)

Theoremblocnilem 22225 Lemma for blocni 22226 and lnocni 22227. 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 22226 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 22227 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 22228 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 22229 A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)

Theoremajfval 22230* The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhmoval 22231* 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 22232 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 22233 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).

Definitiondf-ph 22234* 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 22235 Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphrel 22236 The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphnvi 22237 Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremisphg 22238* 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 22239 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 22240 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 22241 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremelimphu 22242 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 22243* The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
CV

Theoremphpar2 22244 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremphpar 22245 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CV

Theoremip0i 22246 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 22247 Lemma for ip1i 22248. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
CV

Theoremip1i 22248 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremip2i 22249 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdirilem 22250 Lemma for ipdiri 22251. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdiri 22251 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem1 22252 Lemma for ipassi 22262. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem2 22253 Lemma for ipassi 22262. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem3 22254 Lemma for ipassi 22262. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem4 22255 Lemma for ipassi 22262. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem5 22256 Lemma for ipassi 22262. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem7 22257* Lemma for ipassi 22262. Show that is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
fld

Theoremipasslem8 22258* Lemma for ipassi 22262. By ipasslem5 22256, is 0 for all ; since it is continuous and is dense in by qdensere2 18767, we conclude is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)

Theoremipasslem9 22259 Lemma for ipassi 22262. Conclude from ipasslem8 22258 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)

Theoremipasslem10 22260 Lemma for ipassi 22262. Show the inner product associative law for the imaginary number . (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
CV

Theoremipasslem11 22261 Lemma for ipassi 22262. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremipassi 22262 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdir 22263 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipdi 22264 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremip2dii 22265 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)

Theoremdipass 22266 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

Theoremdipassr 22267 "Associative" law for second argument of inner product (compare dipass 22266). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)

Theoremdipassr2 22268 "Associative" law for inner product. Conjugate version of dipassr 22267. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdir 22269 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremdipsubdi 22270 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theorempythi 22271 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 22272 Lemma for sii 22275. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiilem2 22273 Lemma for sii 22275. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
CV

Theoremsiii 22274 Inference from sii 22275. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
CV

Theoremsii 22275 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 22542, bcsiALT 22601, bcsiHIL 22602, csbrn 26308. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
CV

Theoremsspph 22276 A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Theoremipblnfi 22277* 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 22278* 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 22279* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajmoi 22280* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfuni 22281 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Theoremajfun 22282 The adjoint function is a function. This is not immediately apparent from df-aj 22171 but results from the uniqueness shown by ajmoi 22280. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Theoremajval 22283* 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 22284 Extend class notation with the class of all complex Banach spaces.

Definitiondf-cbn 22285 Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)

Theoremiscbn 22286 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 22287 The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theorembnnv 22288 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnrel 22289 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theorembnsscmcl 22290 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 22291 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 22292* Lemma for ubth 22295. 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 19222, 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 22293* Lemma for ubth 22295. 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 22294* Lemma for ubth 22295. Prove the reverse implication, using nmblolbi 22221. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
CV

Theoremubth 22295* 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 22296* Lemma for minveco 22306. 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 22297* Lemma for minveco 22306. 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 22298* Lemma for minveco 22306. 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 22299* Lemma for minveco 22306. is convergent in the subspace topology on . (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
CV

Theoremminvecolem4b 22300* Lemma for minveco 22306. 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

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