HomeHome Metamath Proof Explorer
Theorem List (p. 213 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21494)
  Hilbert Space Explorer  Hilbert Space Explorer
(21495-23017)
  Users' Mathboxes  Users' Mathboxes
(23018-31433)
 

Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnvnnncan2 21201 Cancellation law for vector subtraction. (nnncan2 9080 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A M C ) M ( B M C ) )  =  ( A M B ) )
 
Theoremnvmdi 21202 Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B M C ) )  =  ( ( A S B ) M ( A S C ) ) )
 
Theoremnvnegneg 21203 Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( -u 1 S (
 -u 1 S A ) )  =  A )
 
Theoremnvmul0or 21204 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( ( A S B )  =  Z  <->  ( A  =  0  \/  B  =  Z ) ) )
 
Theoremnvrinv 21205 A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G (
 -u 1 S A ) )  =  Z )
 
Theoremnvlinv 21206 Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( ( -u 1 S A ) G A )  =  Z )
 
Theoremnvsubadd 21207 Relationship between vector subtraction and addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A M B )  =  C  <->  ( B G C )  =  A ) )
 
Theoremnvpncan2 21208 Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A G B ) M A )  =  B )
 
Theoremnvpncan 21209 Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A G B ) M B )  =  A )
 
Theoremnvaddsubass 21210 Associative-type law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) M C )  =  ( A G ( B M C ) ) )
 
Theoremnvaddsub 21211 Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) M C )  =  ( ( A M C ) G B ) )
 
Theoremnvnpcan 21212 Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A M B ) G B )  =  A )
 
Theoremnvaddsub4 21213 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A G B ) M ( C G D ) )  =  ( ( A M C ) G ( B M D ) ) )
 
Theoremnvsubsub23 21214 Swap subtrahend and result of vector subtraction. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A M B )  =  C  <->  ( A M C )  =  B ) )
 
Theoremnvnncan 21215 Cancellation law for a normed complex vector space. (Contributed by NM, 17-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  B )
 
Theoremnvmeq0 21216 The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A M B )  =  Z  <->  A  =  B ) )
 
Theoremnvmid 21217 A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( A M A )  =  Z )
 
Theoremnvf 21218 Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  N : X --> RR )
 
Theoremnvcl 21219 The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( N `  A )  e.  RR )
 
Theoremnvcli 21220 The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  U  e.  NrmCVec   &    |-  A  e.  X   =>    |-  ( N `  A )  e.  RR
 
Theoremnvdm 21221 Two ways to express the set of vectors in a normed complex vector space. (Contributed by NM, 31-Jan-2007.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  ( X  =  dom  N  <->  X  =  ran  G ) )
 
Theoremnvs 21222 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( N `  ( A S B ) )  =  ( ( abs `  A )  x.  ( N `  B ) ) )
 
Theoremnvsge0 21223 The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  RR  /\  0  <_  A )  /\  B  e.  X ) 
 ->  ( N `  ( A S B ) )  =  ( A  x.  ( N `  B ) ) )
 
Theoremnvm1 21224 The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( N `  ( -u 1 S A ) )  =  ( N `
  A ) )
 
Theoremnvdif 21225 The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( -u 1 S B ) ) )  =  ( N `
  ( B G ( -u 1 S A ) ) ) )
 
Theoremnvpi 21226 The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G ( _i S B ) ) )  =  ( N `
  ( B G ( -u _i S A ) ) ) )
 
Theoremnvsub 21227 The norm of the difference between two vectors. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) )  =  ( N `  ( B M A ) ) )
 
Theoremnvz0 21228 The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
 |-  Z  =  ( 0vec `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  ( N `  Z )  =  0 )
 
Theoremnvz 21229 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( ( N `  A )  =  0  <->  A  =  Z ) )
 
Theoremnvtri 21230 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) ) 
 <_  ( ( N `  A )  +  ( N `  B ) ) )
 
Theoremnvmtri 21231 Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A M B ) ) 
 <_  ( ( N `  A )  +  ( N `  B ) ) )
 
Theoremnvmtri2 21232 Triangle inequality for the norm of a vector difference. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( N `  ( A M C ) )  <_  ( ( N `  ( A M B ) )  +  ( N `  ( B M C ) ) ) )
 
Theoremnvabs 21233 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( abs `  (
 ( N `  A )  -  ( N `  B ) ) ) 
 <_  ( N `  ( A G ( -u 1 S B ) ) ) )
 
Theoremnvge0 21234 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  0  <_  ( N `
  A ) )
 
Theoremnvgt0 21235 A nonzero norm is positive. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A  =/=  Z  <->  0  <  ( N `  A ) ) )
 
Theoremnv1 21236 From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/=  Z )  ->  ( N `  ( ( 1  /  ( N `
  A ) ) S A ) )  =  1 )
 
Theoremnvop 21237 A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
 
Theoremnvoprne 21238 The vector addition and scalar product operations are not identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  G  =/=  S )
 
16.2.2  Examples of normed complex vector spaces
 
Theoremcnnv 21239 The set of complex numbers is a normed complex vector space. The vector operation is  +, the scalar product is  x., and the norm function is  abs. (Contributed by Steve Rodriguez, 3-Dec-2006.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  U  e.  NrmCVec
 
Theoremcnnvg 21240 The vector addition (group) operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  +  =  ( +v `  U )
 
Theoremcnnvba 21241 The base set of the normed complex vector space of complex numbers. (Contributed by NM, 7-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  CC  =  (
 BaseSet `  U )
 
Theoremcnnvdemo 21242 Derive the associative law for complex number addition addass 8820 to demonstrate the use of cnnv 21239, cnnvg 21240, and cnnvba 21241. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Theoremcnnvs 21243 The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  x.  =  ( .s OLD `  U )
 
Theoremcnnvnm 21244 The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  abs  =  ( normCV `  U )
 
Theoremcnnvm 21245 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.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  -  =  ( -v `  U )
 
Theoremelimnv 21246 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  U  e. 
 NrmCVec   =>    |-  if ( A  e.  X ,  A ,  Z )  e.  X
 
Theoremelimnvu 21247 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)
 |- 
 if ( U  e.  NrmCVec ,  U ,  <. <.  +  ,  x.  >. ,  abs >. )  e.  NrmCVec
 
16.2.3  Induced metric of a normed complex vector space
 
Theoremimsval 21248 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.)
 |-  M  =  ( -v
 `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
 
Theoremimsdval 21249 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A M B ) ) )
 
Theoremimsdval2 21250 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A G (
 -u 1 S B ) ) ) )
 
Theoremnvnd 21251 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  =  ( A D Z ) )
 
Theoremimsdf 21252 Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  NrmCVec  ->  D : ( X  X.  X ) --> RR )
 
Theoremimsmetlem 21253 Lemma for imsmet 21254. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( inv `  G )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  D  =  (
 IndMet `  U )   &    |-  U  e. 
 NrmCVec   =>    |-  D  e.  ( Met `  X )
 
Theoremimsmet 21254 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  NrmCVec  ->  D  e.  ( Met `  X ) )
 
Theoremimsxmet 21255 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  NrmCVec  ->  D  e.  ( * Met `  X ) )
 
Theoremnvelbl 21256 Membership of a vector in a ball. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X ) )  ->  ( A  e.  ( P ( ball `  D ) R )  <->  ( N `  ( A M P ) )  <  R ) )
 
Theoremnvelbl2 21257 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  D  =  ( IndMet `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  R  e.  RR+ )  /\  ( P  e.  X  /\  A  e.  X ) )  ->  ( ( P G A )  e.  ( P ( ball `  D ) R )  <->  ( N `  A )  <  R ) )
 
Theoremnvlmcl 21258 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   =>    |-  (
 ( U  e.  NrmCVec  /\  F ( ~~> t `  J ) P ) 
 ->  P  e.  X )
 
Theoremnvlmle 21259* 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  N  =  ( normCV `  U )   &    |-  ( ph  ->  U  e.  NrmCVec )   &    |-  ( ph  ->  F : NN
 --> X )   &    |-  ( ph  ->  F ( ~~> t `  J ) P )   &    |-  ( ph  ->  R  e.  RR )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( N `  ( F `  k ) )  <_  R )   =>    |-  ( ph  ->  ( N `  P )  <_  R )
 
Theoremcnims 21260 The metric induced on the complex numbers. cnmet 18277 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by NM, 15-Jan-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   &    |-  D  =  ( abs  o.  -  )   =>    |-  D  =  ( IndMet `  U )
 
Theoremvacn 21261 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.)
 |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  G  =  ( +v `  U )   =>    |-  ( U  e.  NrmCVec  ->  G  e.  ( ( J  tX  J )  Cn  J ) )
 
Theoremnmcvcn 21262 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.)
 |-  N  =  ( normCV `  U )   &    |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  K  =  ( topGen `  ran  (,) )   =>    |-  ( U  e.  NrmCVec  ->  N  e.  ( J  Cn  K ) )
 
Theoremnmcnc 21263 The norm of a normed complex vector space is a continuous function to  CC. (For  RR, see nmcvcn 21262.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.)
 |-  N  =  ( normCV `  U )   &    |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( U  e.  NrmCVec  ->  N  e.  ( J  Cn  K ) )
 
Theoremsmcnlem 21264* Lemma for smcn 21265. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  S  =  ( .s OLD `  U )   &    |-  K  =  ( TopOpen ` fld )   &    |-  X  =  ( BaseSet `  U )   &    |-  N  =  ( normCV `  U )   &    |-  U  e. 
 NrmCVec   &    |-  T  =  ( 1  /  ( 1  +  (
 ( ( ( N `
  y )  +  ( abs `  x )
 )  +  1 ) 
 /  r ) ) )   =>    |-  S  e.  ( ( K  tX  J )  Cn  J )
 
Theoremsmcn 21265 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.)
 |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  S  =  ( .s OLD `  U )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( U  e.  NrmCVec  ->  S  e.  ( ( K  tX  J )  Cn  J ) )
 
Theoremvmcn 21266 Vector subtraction is jointly continuous in both arguments. (Contributed by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  M  =  ( -v `  U )   =>    |-  ( U  e.  NrmCVec  ->  M  e.  ( ( J  tX  J )  Cn  J ) )
 
16.2.4  Inner product
 
Syntaxcdip 21267 Extend class notation with the class inner product functions.
 class  .i OLD
 
Definitiondf-dip 21268* 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  ( 1st `  w
), the scalar product is  ( 2nd `  w
), and the norm is  n. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
 |- 
 .i OLD  =  ( u  e.  NrmCVec  |->  ( x  e.  ( BaseSet `  u ) ,  y  e.  ( BaseSet `  u )  |->  ( sum_ k  e.  (
 1 ... 4 ) ( ( _i ^ k
 )  x.  ( ( ( normCV `  u ) `  ( x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  /  4 ) ) )
 
Theoremdipfval 21269* 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( U  e.  NrmCVec  ->  P  =  ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4
 ) ( ( _i
 ^ k )  x.  ( ( N `  ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) )  /  4 ) ) )
 
Theoremipval 21270* 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  G, the scalar product is  S, the norm is  N, and the set of vectors is  X. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k )  x.  ( ( N `
  ( A G ( ( _i ^
 k ) S B ) ) ) ^
 2 ) )  / 
 4 ) )
 
Theoremipval2lem2 21271 Lemma for ipval3 21276. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( ( N `
  ( A G ( C S B ) ) ) ^ 2
 )  e.  RR )
 
Theoremipval2lem3 21272 Lemma for ipval3 21276. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( N `  ( A G B ) ) ^ 2 )  e.  RR )
 
Theoremipval2lem4 21273 Lemma for ipval3 21276. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( ( N `
  ( A G ( C S B ) ) ) ^ 2
 )  e.  CC )
 
Theoremipval2 21274 Expansion of the inner product value ipval 21270. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( (
 ( ( ( N `
  ( A G B ) ) ^
 2 )  -  (
 ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `
  ( A G ( _i S B ) ) ) ^ 2
 )  -  ( ( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) )  /  4
 ) )
 
Theorem4ipval2 21275 Four times the inner product value ipval3 21276, useful for simplifying certain proofs. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( 4  x.  ( A P B ) )  =  ( ( ( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  +  ( _i 
 x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) ) )
 
Theoremipval3 21276 Expansion of the inner product value ipval 21270. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  M  =  ( -v
 `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  (
 ( ( ( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `  ( A M B ) ) ^ 2 ) )  +  ( _i 
 x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) )  / 
 4 ) )
 
Theoremipval2lem5 21277 Lemma for ipval3 21276. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  M  =  ( -v
 `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( ( N `  ( A M ( C S B ) ) ) ^ 2 )  e. 
 RR )
 
Theoremipval2lem6 21278 Lemma for ipval3 21276. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  M  =  ( -v
 `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( N `
  ( A M B ) ) ^
 2 )  e.  RR )
 
Theorem4ipval3 21279 Four times the inner product value ipval3 21276, useful for simplifying certain proofs. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  M  =  ( -v
 `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( 4  x.  ( A P B ) )  =  (
 ( ( ( N `
  ( A G B ) ) ^
 2 )  -  (
 ( N `  ( A M B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `
  ( A G ( _i S B ) ) ) ^ 2
 )  -  ( ( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) ) )
 
Theoremipidsq 21280 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( A P A )  =  ( ( N `  A ) ^
 2 ) )
 
Theoremipnm 21281 Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( N `  A )  =  ( sqr `  ( A P A ) ) )
 
Theoremdipcl 21282 An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
 
Theoremipf 21283 Mapping for the inner product operation. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( U  e.  NrmCVec  ->  P : ( X  X.  X ) --> CC )
 
Theoremdipcj 21284 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( * `  ( A P B ) )  =  ( B P A ) )
 
Theoremipipcj 21285 An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A P B )  x.  ( B P A ) )  =  ( ( abs `  ( A P B ) ) ^ 2
 ) )
 
Theoremdiporthcom 21286 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A P B )  =  0  <->  ( B P A )  =  0 ) )
 
Theoremdip0r 21287 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( A P Z )  =  0 )
 
Theoremdip0l 21288 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( Z P A )  =  0 )
 
Theoremipz 21289 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.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( ( A P A )  =  0  <->  A  =  Z ) )
 
Theoremdipcn 21290 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.)
 |-  P  =  ( .i
 OLD `  U )   &    |-  C  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( U  e.  NrmCVec  ->  P  e.  ( ( J  tX  J )  Cn  K ) )
 
16.2.5  Subspaces
 
Syntaxcss 21291 Extend class notation with the class of all subspaces of complex normed vector spaces.
 class  SubSp
 
Definitiondf-ssp 21292* Define the class of all subspaces of complex normed vector spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |- 
 SubSp  =  ( u  e. 
 NrmCVec 
 |->  { w  e.  NrmCVec  |  ( ( +v `  w )  C_  ( +v
 `  u )  /\  ( .s OLD `  w )  C_  ( .s OLD `  u )  /\  ( normCV `  w )  C_  ( normCV `  u ) ) }
 )
 
Theoremsspval 21293* 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.)
 |-  G  =  ( +v
 `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  H  =  ( SubSp `  U )   =>    |-  ( U  e.  NrmCVec  ->  H  =  { w  e. 
 NrmCVec  |  ( ( +v
 `  w )  C_  G  /\  ( .s OLD `  w )  C_  S  /\  ( normCV `  w )  C_  N ) } )
 
Theoremisssp 21294 The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  F  =  ( +v `  W )   &    |-  S  =  ( .s
 OLD `  U )   &    |-  R  =  ( .s OLD `  W )   &    |-  N  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec  /\  ( F  C_  G  /\  R  C_  S  /\  M  C_  N ) ) ) )
 
Theoremsspid 21295 A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  H  =  ( SubSp `  U )   =>    |-  ( U  e.  NrmCVec  ->  U  e.  H )
 
Theoremsspnv 21296 A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
 |-  H  =  ( SubSp `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  H ) 
 ->  W  e.  NrmCVec )
 
Theoremsspba 21297 The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  X )
 
Theoremsspg 21298 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.)
 |-  Y  =  ( BaseSet `  W )   &    |-  G  =  ( +v `  U )   &    |-  F  =  ( +v `  W )   &    |-  H  =  (
 SubSp `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  F  =  ( G  |`  ( Y  X.  Y ) ) )
 
Theoremsspgval 21299 Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  G  =  ( +v `  U )   &    |-  F  =  ( +v `  W )   &    |-  H  =  (
 SubSp `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y )
 )  ->  ( A F B )  =  ( A G B ) )
 
Theoremssps 21300 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.)
 |-  Y  =  ( BaseSet `  W )   &    |-  S  =  ( .s OLD `  U )   &    |-  R  =  ( .s
 OLD `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  =  ( S  |`  ( CC  X.  Y ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31433
  Copyright terms: Public domain < Previous  Next >