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Definition df-aj 21442
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 21440 . 2  class  adj
2 vu . . 3  set  u
3 vw . . 3  set  w
4 cnv 21254 . . 3  class  NrmCVec
52cv 1641 . . . . . . 7  class  u
6 cba 21256 . . . . . . 7  class  BaseSet
75, 6cfv 5337 . . . . . 6  class  ( BaseSet `  u )
83cv 1641 . . . . . . 7  class  w
98, 6cfv 5337 . . . . . 6  class  ( BaseSet `  w )
10 vt . . . . . . 7  set  t
1110cv 1641 . . . . . 6  class  t
127, 9, 11wf 5333 . . . . 5  wff  t : ( BaseSet `  u ) --> ( BaseSet `  w )
13 vs . . . . . . 7  set  s
1413cv 1641 . . . . . 6  class  s
159, 7, 14wf 5333 . . . . 5  wff  s : ( BaseSet `  w ) --> ( BaseSet `  u )
16 vx . . . . . . . . . . 11  set  x
1716cv 1641 . . . . . . . . . 10  class  x
1817, 11cfv 5337 . . . . . . . . 9  class  ( t `
 x )
19 vy . . . . . . . . . 10  set  y
2019cv 1641 . . . . . . . . 9  class  y
21 cdip 21387 . . . . . . . . . 10  class  .i OLD
228, 21cfv 5337 . . . . . . . . 9  class  ( .i
OLD `  w )
2318, 20, 22co 5945 . . . . . . . 8  class  ( ( t `  x ) ( .i OLD `  w
) y )
2420, 14cfv 5337 . . . . . . . . 9  class  ( s `
 y )
255, 21cfv 5337 . . . . . . . . 9  class  ( .i
OLD `  u )
2617, 24, 25co 5945 . . . . . . . 8  class  ( x ( .i OLD `  u
) ( s `  y ) )
2723, 26wceq 1642 . . . . . . 7  wff  ( ( t `  x ) ( .i OLD `  w
) y )  =  ( x ( .i
OLD `  u )
( s `  y
) )
2827, 19, 9wral 2619 . . . . . 6  wff  A. y  e.  ( BaseSet `  w )
( ( t `  x ) ( .i
OLD `  w )
y )  =  ( x ( .i OLD `  u ) ( s `
 y ) )
2928, 16, 7wral 2619 . . . . 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 934 . . . 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 4157 . . 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 5947 . 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 1642 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  21501
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