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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsspimsval 22101 The induced metric on a subspace in terms of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Theoremsspims 22102 The induced metric on a subspace is a restriction of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

17.3  Operators on complex vector spaces

17.3.1  Definitions and basic properties

Syntaxclno 22103 Extend class notation with the class of linear operators on normed complex vector spaces.

Syntaxcnmoo 22104 Extend class notation with the class of operator norms on normed complex vector spaces.

Syntaxcblo 22105 Extend class notation with the class of bounded linear operators on normed complex vector spaces.

Syntaxc0o 22106 Extend class notation with the class of zero operators on normed complex vector spaces.

Definitiondf-lno 22107* Define the class of linear operators between two normed complex vector spaces. In the literature, an operator may be a partial function, i.e. the domain of an operator is not necessarily the entire vector space. However, since the domain of a linear operator is a vector subspace, we define it with a complete function for convenience and will use subset relations to specify the partial function case. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)

Definitiondf-nmoo 22108* Define the norm of an operator between two normed complex vector spaces. This definition produces an operator norm function for each pair of vector spaces . Based on definition of linear operator norm in [AkhiezerGlazman] p. 39, although we define it for all operators for convenience. It isn't necessarily meaningful for nonlinear operators, since it doesn't take into account operator values at vectors with norm greater than 1. See Equation 2 of [Kreyszig] p. 92 for a definition that does (although it ignores the value at the zero vector). However, operator norms are rarely if ever used for nonlinear operators. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
CV CV

Definitiondf-blo 22109* Define the class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)

Definitiondf-0o 22110* Define the zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)

Syntaxcaj 22111 Adjoint of an operator.

Syntaxchmo 22112 Set of Hermitional (self-adjoint) operators.

Definitiondf-aj 22113* Define the adjoint of an operator (if it exists). The domain of is the set of all operators from to that have an adjoint. Definition 3.9-1 of [Kreyszig] p. 196, although we don't require that and be Hilbert spaces nor that the operators be linear. Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

Definitiondf-hmo 22114* Define the set of Hermitian (self-adjoint) operators on a normed complex vector space (normally a Hilbert space). Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Theoremlnoval 22115* The set of 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.)

Theoremislno 22116* The predicate "is a linear operator." (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremlnolin 22117 Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremlnof 22118 A linear operator is a mapping. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)

Theoremlno0 22119 The value of a linear operator at zero is zero. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)

Theoremlnocoi 22120 The composition of two linear operators is linear. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremlnoadd 22121 Addition property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremlnosub 22122 Subtraction property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremlnomul 22123 Scalar multiplication property of a linear operator. (Contributed by NM, 5-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremnvo00 22124 Two ways to express a zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)

Theoremnmoofval 22125* The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
CV       CV

Theoremnmooval 22126* The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
CV       CV

Theoremnmosetre 22127* The set in the supremum of the operator norm definition df-nmoo 22108 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
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Theoremnmosetn0 22128* The set in the supremum of the operator norm definition df-nmoo 22108 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
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Theoremnmoxr 22129 The norm of an operator is an extended real. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)

Theoremnmooge0 22130 The norm of an operator is nonnegative. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnmorepnf 22131 The norm of an operator is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnmoreltpnf 22132 The norm of any operator is real iff it is less than plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnmogtmnf 22133 The norm of an operator is greater than minus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnmoolb 22134 A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremnmoubi 22135* An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremnmoub3i 22136* An upper bound for an operator norm. (Contributed by NM, 12-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremnmoub2i 22137* An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremnmobndi 22138* Two ways to express that an operator is bounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremnmounbi 22139* Two ways two express that an operator is unbounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremnmounbseqi 22140* An unbounded operator determines an unbounded sequence. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.)
CV       CV

TheoremnmounbseqiOLD 22141* An unbounded operator determines an unbounded sequence. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
CV       CV

Theoremnmobndseqi 22142* A bounded sequence determines a bounded operator. (Contributed by NM, 18-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.)
CV       CV

TheoremnmobndseqiOLD 22143* A bounded sequence determines a bounded operator. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
CV       CV

Theorembloval 22144* 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 22145 The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)

Theoremisblo2 22146 The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theorembloln 22147 A bounded operator is a linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremblof 22148 A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnmblore 22149 The norm of a bounded operator is a real number. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theorem0ofval 22150 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 22151 Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)

Theorem0oo 22152 The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)

Theorem0lno 22153 The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremnmoo0 22154 The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)

Theorem0blo 22155 The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnmlno0lem 22156 Lemma for nmlno0i 22157. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
CV       CV

Theoremnmlno0i 22157 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)

Theoremnmlno0 22158 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)

Theoremnmlnoubi 22159* 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 22160 The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)

Theoremlnon0 22161* The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)

Theoremnmblolbii 22162 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremnmblolbi 22163 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
CV       CV

Theoremisblo3i 22164* 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 22165* Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremblometi 22166 Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)

Theoremblocnilem 22167 Lemma for blocni 22168 and lnocni 22169. 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 22168 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 22169 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 22170 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 22171 A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)

Theoremajfval 22172* The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhmoval 22173* 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 22174 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 22175 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).

Definitiondf-ph 22176* 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 22177 Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphrel 22178 The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Theoremphnvi 22179 Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Theoremisphg 22180* 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 22181 A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
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17.4.2  Examples of pre-Hilbert spaces

Theoremcncph 22182 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 22183 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremelimphu 22184 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 22185* The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
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Theoremphpar2 22186 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
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Theoremphpar 22187 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
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Theoremip0i 22188 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.)
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Theoremip1ilem 22189 Lemma for ip1i 22190. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
CV

Theoremip1i 22190 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremip2i 22191 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdirilem 22192 Lemma for ipdiri 22193. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremipdiri 22193 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem1 22194 Lemma for ipassi 22204. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem2 22195 Lemma for ipassi 22204. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem3 22196 Lemma for ipassi 22204. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem4 22197 Lemma for ipassi 22204. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem5 22198 Lemma for ipassi 22204. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Theoremipasslem7 22199* Lemma for ipassi 22204. Show that is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
fld

Theoremipasslem8 22200* Lemma for ipassi 22204. By ipasslem5 22198, is 0 for all ; since it is continuous and is dense in by qdensere2 18713, we conclude is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)

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