MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-aj Unicode version

Definition df-aj 21320
Description: Define the adjoint of an operator (if it exists). The domain of  U adj W is the set of all operators from  U to  W that have an adjoint. Definition 3.9-1 of [Kreyszig] p. 196, although we don't require that  U and  W 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.)
Assertion
Ref Expression
df-aj  |-  adj  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  { <. t ,  s >.  |  ( t : ( BaseSet `  u ) --> ( BaseSet `  w )  /\  s : ( BaseSet `  w
) --> ( BaseSet `  u
)  /\  A. x  e.  ( BaseSet `  u ) A. y  e.  ( BaseSet
`  w ) ( ( t `  x
) ( .i OLD `  w ) y )  =  ( x ( .i OLD `  u
) ( s `  y ) ) ) } )
Distinct variable group:    t, s, u, w, x, y

Detailed syntax breakdown of Definition df-aj
StepHypRef Expression
1 caj 21318 . 2  class  adj
2 vu . . 3  set  u
3 vw . . 3  set  w
4 cnv 21132 . . 3  class  NrmCVec
52cv 1627 . . . . . . 7  class  u
6 cba 21134 . . . . . . 7  class  BaseSet
75, 6cfv 5221 . . . . . 6  class  ( BaseSet `  u )
83cv 1627 . . . . . . 7  class  w
98, 6cfv 5221 . . . . . 6  class  ( BaseSet `  w )
10 vt . . . . . . 7  set  t
1110cv 1627 . . . . . 6  class  t
127, 9, 11wf 5217 . . . . 5  wff  t : ( BaseSet `  u ) --> ( BaseSet `  w )
13 vs . . . . . . 7  set  s
1413cv 1627 . . . . . 6  class  s
159, 7, 14wf 5217 . . . . 5  wff  s : ( BaseSet `  w ) --> ( BaseSet `  u )
16 vx . . . . . . . . . . 11  set  x
1716cv 1627 . . . . . . . . . 10  class  x
1817, 11cfv 5221 . . . . . . . . 9  class  ( t `
 x )
19 vy . . . . . . . . . 10  set  y
2019cv 1627 . . . . . . . . 9  class  y
21 cdip 21265 . . . . . . . . . 10  class  .i OLD
228, 21cfv 5221 . . . . . . . . 9  class  ( .i
OLD `  w )
2318, 20, 22co 5819 . . . . . . . 8  class  ( ( t `  x ) ( .i OLD `  w
) y )
2420, 14cfv 5221 . . . . . . . . 9  class  ( s `
 y )
255, 21cfv 5221 . . . . . . . . 9  class  ( .i
OLD `  u )
2617, 24, 25co 5819 . . . . . . . 8  class  ( x ( .i OLD `  u
) ( s `  y ) )
2723, 26wceq 1628 . . . . . . 7  wff  ( ( t `  x ) ( .i OLD `  w
) y )  =  ( x ( .i
OLD `  u )
( s `  y
) )
2827, 19, 9wral 2544 . . . . . 6  wff  A. y  e.  ( BaseSet `  w )
( ( t `  x ) ( .i
OLD `  w )
y )  =  ( x ( .i OLD `  u ) ( s `
 y ) )
2928, 16, 7wral 2544 . . . . 5  wff  A. x  e.  ( BaseSet `  u ) A. y  e.  ( BaseSet
`  w ) ( ( t `  x
) ( .i OLD `  w ) y )  =  ( x ( .i OLD `  u
) ( s `  y ) )
3012, 15, 29w3a 939 . . . 4  wff  ( t : ( BaseSet `  u
) --> ( BaseSet `  w
)  /\  s :
( BaseSet `  w ) --> ( BaseSet `  u )  /\  A. x  e.  (
BaseSet `  u ) A. y  e.  ( BaseSet `  w ) ( ( t `  x ) ( .i OLD `  w
) y )  =  ( x ( .i
OLD `  u )
( s `  y
) ) )
3130, 10, 13copab 4077 . . 3  class  { <. t ,  s >.  |  ( t : ( BaseSet `  u ) --> ( BaseSet `  w )  /\  s : ( BaseSet `  w
) --> ( BaseSet `  u
)  /\  A. x  e.  ( BaseSet `  u ) A. y  e.  ( BaseSet
`  w ) ( ( t `  x
) ( .i OLD `  w ) y )  =  ( x ( .i OLD `  u
) ( s `  y ) ) ) }
322, 3, 4, 4, 31cmpt2 5821 . 2  class  ( u  e.  NrmCVec ,  w  e.  NrmCVec 
|->  { <. t ,  s
>.  |  ( t : ( BaseSet `  u
) --> ( BaseSet `  w
)  /\  s :
( BaseSet `  w ) --> ( BaseSet `  u )  /\  A. x  e.  (
BaseSet `  u ) A. y  e.  ( BaseSet `  w ) ( ( t `  x ) ( .i OLD `  w
) y )  =  ( x ( .i
OLD `  u )
( s `  y
) ) ) } )
331, 32wceq 1628 1  wff  adj  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  { <. t ,  s >.  |  ( t : ( BaseSet `  u ) --> ( BaseSet `  w )  /\  s : ( BaseSet `  w
) --> ( BaseSet `  u
)  /\  A. x  e.  ( BaseSet `  u ) A. y  e.  ( BaseSet
`  w ) ( ( t `  x
) ( .i OLD `  w ) y )  =  ( x ( .i OLD `  u
) ( s `  y ) ) ) } )
Colors of variables: wff set class
This definition is referenced by:  ajfval  21379
  Copyright terms: Public domain W3C validator