HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  df-oc Structured version   Visualization version   Unicode version

Definition df-oc 26954
Description: Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocval 26982 and chocvali 27001 for its value. Textbooks usually denote this unary operation with the symbol  _|_ as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation)  _|_ rather than introducing a new syntactic form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
Assertion
Ref Expression
df-oc  |-  _|_  =  ( x  e.  ~P ~H  |->  { y  e. 
~H  |  A. z  e.  x  ( y  .ih  z )  =  0 } )
Distinct variable group:    x, y, z

Detailed syntax breakdown of Definition df-oc
StepHypRef Expression
1 cort 26632 . 2  class  _|_
2 vx . . 3  setvar  x
3 chil 26621 . . . 4  class  ~H
43cpw 3963 . . 3  class  ~P ~H
5 vy . . . . . . . 8  setvar  y
65cv 1454 . . . . . . 7  class  y
7 vz . . . . . . . 8  setvar  z
87cv 1454 . . . . . . 7  class  z
9 csp 26624 . . . . . . 7  class  .ih
106, 8, 9co 6315 . . . . . 6  class  ( y 
.ih  z )
11 cc0 9565 . . . . . 6  class  0
1210, 11wceq 1455 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1454 . . . . 5  class  x
1412, 7, 13wral 2749 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2753 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4475 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1455 1  wff  _|_  =  ( x  e.  ~P ~H  |->  { y  e. 
~H  |  A. z  e.  x  ( y  .ih  z )  =  0 } )
Colors of variables: wff setvar class
This definition is referenced by:  ocval  26982
  Copyright terms: Public domain W3C validator