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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnvaddsubass 22001 Associative-type law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsub 22002 Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvnpcan 22003 Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsub4 22004 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)

Theoremnvsubsub23 22005 Swap subtrahend and result of vector subtraction. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvnncan 22006 Cancellation law for a normed complex vector space. (Contributed by NM, 17-Dec-2007.) (New usage is discouraged.)

Theoremnvmeq0 22007 The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvmid 22008 A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvf 22009 Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvcl 22010 The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvcli 22011 The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.)
CV

Theoremnvdm 22012 Two ways to express the set of vectors in a normed complex vector space. (Contributed by NM, 31-Jan-2007.) (New usage is discouraged.)
CV

Theoremnvs 22013 Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvsge0 22014 The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
CV

Theoremnvm1 22015 The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvdif 22016 The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.)
CV

Theoremnvpi 22017 The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
CV

Theoremnvsub 22018 The norm of the difference between two vectors. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
CV

Theoremnvz0 22019 The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvz 22020 The norm of a vector is zero iff the vector is zero. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvtri 22021 Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
CV

Theoremnvmtri 22022 Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
CV

Theoremnvmtri2 22023 Triangle inequality for the norm of a vector difference. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
CV

Theoremnvabs 22024 Norm difference property of a normed complex vector space. Problem 3 of [Kreyszig] p. 64. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
CV

Theoremnvge0 22025 The norm of a normed complex vector space is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvgt0 22026 A nonzero norm is positive. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
CV

Theoremnv1 22027 From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
CV

Theoremnvop 22028 A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
CV

Theoremnvoprne 22029 The vector addition and scalar product operations are not identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

17.2.2  Examples of normed complex vector spaces

Theoremcnnv 22030 The set of complex numbers is a normed complex vector space. The vector operation is , the scalar product is , and the norm function is . (Contributed by Steve Rodriguez, 3-Dec-2006.) (New usage is discouraged.)

Theoremcnnvg 22031 The vector addition (group) operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)

Theoremcnnvba 22032 The base set of the normed complex vector space of complex numbers. (Contributed by NM, 7-Nov-2007.) (New usage is discouraged.)

Theoremcnnvdemo 22033 Derive the associative law for complex number addition addass 9024 to demonstrate the use of cnnv 22030, cnnvg 22031, and cnnvba 22032. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcnnvs 22034 The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)

Theoremcnnvnm 22035 The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
CV

Theoremcnnvm 22036 The vector subtraction operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremelimnv 22037 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)

Theoremelimnvu 22038 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)

17.2.3  Induced metric of a normed complex vector space

Theoremimsval 22039 Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
CV

Theoremimsdval 22040 Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
CV

Theoremimsdval2 22041 Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvnd 22042 The norm of a normed complex vector space expressed in terms of the distance function of its induced metric. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
CV

Theoremimsdf 22043 Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)

Theoremimsmetlem 22044 Lemma for imsmet 22045. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
CV

Theoremimsmet 22045 The induced metric of a normed complex vector space is a metric space. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremimsxmet 22046 The induced metric of a normed complex vector space is an extended metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremnvelbl 22047 Membership of a vector in a ball. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
CV

Theoremnvelbl2 22048 Membership of an off-center vector in a ball. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
CV

Theoremnvlmcl 22049 Closure of the limit of a converging vector sequence. (Contributed by NM, 26-Dec-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremnvlmle 22050* If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value. (Contributed by NM, 25-Dec-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
CV

Theoremcnims 22051 The metric induced on the complex numbers. cnmet 18691 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by NM, 15-Jan-2008.) (New usage is discouraged.)

Theoremvacn 22052 Vector addition is jointly continuous in both arguments. (Contributed by Jeffrey Hankins, 16-Jun-2009.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremnmcvcn 22053 The norm of a normed complex vector space is a continuous function. (Contributed by NM, 16-May-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
CV

Theoremnmcnc 22054 The norm of a normed complex vector space is a continuous function to . (For , see nmcvcn 22053.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.)
CV                     fld

Theoremsmcnlem 22055* Lemma for smcn 22056. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
fld              CV

Theoremsmcn 22056 Scalar multiplication is jointly continuous in both arguments. (Contributed by NM, 16-Jun-2009.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
fld

Theoremvmcn 22057 Vector subtraction is jointly continuous in both arguments. (Contributed by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)

17.2.4  Inner product

Syntaxcdip 22058 Extend class notation with the class inner product functions.

Definitiondf-dip 22059* Define a function that maps a complex normed vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is , the scalar product is , and the norm is . (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
CV

Theoremdipfval 22060* The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law. (Contributed by NM, 10-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
CV

Theoremipval 22061* Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is , the scalar product is , the norm is , and the set of vectors is . Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
CV

Theoremipval2lem2 22062 Lemma for ipval3 22067. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
CV

Theoremipval2lem3 22063 Lemma for ipval3 22067. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
CV

Theoremipval2lem4 22064 Lemma for ipval3 22067. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
CV

Theoremipval2 22065 Expansion of the inner product value ipval 22061. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
CV

Theorem4ipval2 22066 Four times the inner product value ipval3 22067, useful for simplifying certain proofs. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
CV

Theoremipval3 22067 Expansion of the inner product value ipval 22061. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
CV

Theoremipval2lem5 22068 Lemma for ipval3 22067. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
CV

Theoremipval2lem6 22069 Lemma for ipval3 22067. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
CV

Theorem4ipval3 22070 Four times the inner product value ipval3 22067, useful for simplifying certain proofs. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
CV

Theoremipidsq 22071 The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
CV

Theoremipnm 22072 Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
CV

Theoremdipcl 22073 An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)

Theoremipf 22074 Mapping for the inner product operation. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremdipcj 22075 The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremipipcj 22076 An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)

Theoremdiporthcom 22077 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)

Theoremdip0r 22078 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.)

Theoremdip0l 22079 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.)

Theoremipz 22080 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremdipcn 22081 Inner product is jointly continuous in both arguments. (Contributed by NM, 21-Aug-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
fld

17.2.5  Subspaces

Syntaxcss 22082 Extend class notation with the class of all subspaces of complex normed vector spaces.

Definitiondf-ssp 22083* Define the class of all subspaces of complex normed vector spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
CV CV

Theoremsspval 22084* The set of all subspaces of a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
CV              CV

Theoremisssp 22085 The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremsspid 22086 A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremsspnv 22087 A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)

Theoremsspba 22088 The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)

Theoremsspg 22089 Vector addition on a subspace is a restriction of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremsspgval 22090 Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremssps 22091 Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremsspsval 22092 Scalar multiplication on a subspace in terms of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremsspmlem 22093* Lemma for sspm 22095 and others. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Theoremsspmval 22094 Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremsspm 22095 Vector subtraction on a subspace is a restriction of vector subtraction on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremsspz 22096 The zero vector of a subspace is the same as the parent's. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremsspn 22097 The norm on a subspace is a restriction of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremsspnval 22098 The norm on a subspace in terms of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
CV       CV

Theoremsspival 22099 The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremsspi 22100 The inner product on a subspace is a restriction of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

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