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Definition df-aj 21274
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 21272 . 2  class  adj
2 vu . . 3  set  u
3 vw . . 3  set  w
4 cnv 21086 . . 3  class  NrmCVec
52cv 1618 . . . . . . 7  class  u
6 cba 21088 . . . . . . 7  class  BaseSet
75, 6cfv 4659 . . . . . 6  class  ( BaseSet `  u )
83cv 1618 . . . . . . 7  class  w
98, 6cfv 4659 . . . . . 6  class  ( BaseSet `  w )
10 vt . . . . . . 7  set  t
1110cv 1618 . . . . . 6  class  t
127, 9, 11wf 4655 . . . . 5  wff  t : ( BaseSet `  u ) --> ( BaseSet `  w )
13 vs . . . . . . 7  set  s
1413cv 1618 . . . . . 6  class  s
159, 7, 14wf 4655 . . . . 5  wff  s : ( BaseSet `  w ) --> ( BaseSet `  u )
16 vx . . . . . . . . . . 11  set  x
1716cv 1618 . . . . . . . . . 10  class  x
1817, 11cfv 4659 . . . . . . . . 9  class  ( t `
 x )
19 vy . . . . . . . . . 10  set  y
2019cv 1618 . . . . . . . . 9  class  y
21 cdip 21219 . . . . . . . . . 10  class  .i OLD
228, 21cfv 4659 . . . . . . . . 9  class  ( .i
OLD `  w )
2318, 20, 22co 5778 . . . . . . . 8  class  ( ( t `  x ) ( .i OLD `  w
) y )
2420, 14cfv 4659 . . . . . . . . 9  class  ( s `
 y )
255, 21cfv 4659 . . . . . . . . 9  class  ( .i
OLD `  u )
2617, 24, 25co 5778 . . . . . . . 8  class  ( x ( .i OLD `  u
) ( s `  y ) )
2723, 26wceq 1619 . . . . . . 7  wff  ( ( t `  x ) ( .i OLD `  w
) y )  =  ( x ( .i
OLD `  u )
( s `  y
) )
2827, 19, 9wral 2516 . . . . . 6  wff  A. y  e.  ( BaseSet `  w )
( ( t `  x ) ( .i
OLD `  w )
y )  =  ( x ( .i OLD `  u ) ( s `
 y ) )
2928, 16, 7wral 2516 . . . . 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 4036 . . 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 5780 . 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 1619 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  21333
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