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Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
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PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)

16.1  Complex vector spaces

16.1.1  Definition and basic properties

Syntaxcvc 21101 Extend class notation with the class of all complex vector spaces.

Definitiondf-vc 21102* Define the class of all complex vector spaces. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcrel 21103 The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremvci 21104* The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable was chosen because is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcsm 21105 Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvccl 21106 Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcid 21107 Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcdi 21108 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcdir 21109 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcass 21110 Associative law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvc2 21111 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremvcsubdir 21112 Subtractive distributive law for the scalar product of a complex vector space. (Contributed by NM, 31-Jul-2007.) (New usage is discouraged.)

Theoremvcablo 21113 Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcgrp 21114 Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvcgcl 21115 Closure law for the vector addition (group) operation of a complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremvccom 21116 Vector addition is commutative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvcaass 21117 Vector addition is associative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvca32 21118 Commutative/associative law that swaps the last two terms in a triple vector sum. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremvca4 21119 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremvcrcan 21120 Right cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvclcan 21121 Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvczcl 21122 The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvc0rid 21123 The zero vector is a right identity element. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvc0lid 21124 The zero vector is a left identity element. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
GId

Theoremvc0 21125 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvcz 21126 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
GId

Theoremvcm 21127 Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 25-Nov-2006.) (New usage is discouraged.)

Theoremvcrinv 21128 A vector minus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvclinv 21129 Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvcnegneg 21130 Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcnegsubdi2 21131 Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcsub4 21132 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremisvclem 21133* Lemma for isvc 21137. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprnelem 21134 Lemma for vcoprne 21135. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprne 21135 The operations of a complex vector space cannot be identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcex 21136 The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvc 21137* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvci 21138* Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

16.1.2  Examples of complex vector spaces

Theoremcncvc 21139 The set of complex numbers is a complex vector space. The vector operation is , and the scalar product is . (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

16.2  Normed complex vector spaces

16.2.1  Definition and basic properties

Syntaxcnv 21140 Extend class notation with the class of all normed complex vector spaces.

Syntaxcpv 21141 Extend class notation with vector addition in a normed complex vector space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition caddc 8740.

Syntaxcba 21142 Extend class notation with the base set of a normed complex vector space. (Note that is capitalized because, once it is fixed for a particular vector space , it is not a function, unlike e.g. CV. This is our typical convention.) (New usage is discouraged.)

Syntaxcns 21143 Extend class notation with scalar multiplication in a normed complex vector space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.

Syntaxcn0v 21144 Extend class notation with zero vector in a normed complex vector space.

Syntaxcnsb 21145 Extend class notation with vector subtraction in a normed complex vector space.

Syntaxcnmcv 21146 Extend class notation with the norm function in a normed complex vector space. In the literature, the norm of is usually written "|| ||", but we use function notation to take advantage of our existing theorems about functions.
CV

Syntaxcims 21147 Extend class notation with the class of the induced metrics on normed complex vector spaces.

Definitiondf-nv 21148* Define the class of all normed complex vector spaces. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvss 21149 Structure of the class of all normed complex vectors spaces. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)

Theoremnvvcop 21150 A normed complex vector space is a vector space. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)

Definitiondf-va 21151 Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-ba 21152 Define the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-sm 21153 Define scalar multiplication on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)

Definitiondf-0v 21154 Define the zero vector in a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
GId

Definitiondf-vs 21155 Define vector subtraction on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Definitiondf-nmcv 21156 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Definitiondf-ims 21157 Define the induced metric on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
CV

Theoremnvrel 21158 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

Theoremvafval 21159 Value of the function for the vector addition (group) operation on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Theorembafval 21160 Value of the function for the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremsmfval 21161 Value of the function for the scalar multiplication operation on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)

Theorem0vfval 21162 Value of the function for the zero vector on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
GId

Theoremnmcvfval 21163 Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Theoremnvop2 21164 A normed complex vector space is an ordered pair of a vector space and a norm operation. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvvop 21165 The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)

Theoremisnvlem 21166* Lemma for isnv 21168. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvex 21167 The components of a normed complex vector space are sets. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)

Theoremisnv 21168* The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
GId

Theoremisnvi 21169* Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
GId

Theoremnvi 21170* The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
CV

Theoremnvvc 21171 The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvablo 21172 The vector addition operation of a normed complex vector space is an Abelian group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvgrp 21173 The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvgf 21174 Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvsf 21175 Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvgcl 21176 Closure law for the vector addition (group) operation of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Theoremnvcom 21177 The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvass 21178 The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvadd12 21179 Commutative/associative law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd32 21180 Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

Theoremnvrcan 21181 Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvlcan 21182 Left cancellation law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd4 21183 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)

Theoremnvscl 21184 Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremnvsid 21185 Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvsass 21186 Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremnvscom 21187 Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Theoremnvdi 21188 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvdir 21189 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnv2 21190 A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)

Theoremvsfval 21191 Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)

Theoremnvzcl 21192 Closure law for the zero vector of a normed complex vector space. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnv0rid 21193 The zero vector is a right identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnv0lid 21194 The zero vector is a left identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnv0 21195 Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvsz 21196 Anything times the zero vector is the zero vector. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvinv 21197 Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvinvfval 21198 Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.) (Contributed by NM, 27-Mar-2008.) (New usage is discouraged.)

Theoremnvm 21199 Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvmval 21200 Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

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