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Theorem List for Metamath Proof Explorer - 30501-30600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhosubcl 30501 Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ (๐‘† โˆ’op ๐‘‡): โ„‹โŸถ โ„‹)
 
Theoremhoaddcom 30502 Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ (๐‘† +op ๐‘‡) = (๐‘‡ +op ๐‘†))
 
Theoremhodsi 30503 Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ((๐‘… โˆ’op ๐‘†) = ๐‘‡ โ†” (๐‘† +op ๐‘‡) = ๐‘…)
 
Theoremhoaddassi 30504 Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ((๐‘… +op ๐‘†) +op ๐‘‡) = (๐‘… +op (๐‘† +op ๐‘‡))
 
Theoremhoadd12i 30505 Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (๐‘… +op (๐‘† +op ๐‘‡)) = (๐‘† +op (๐‘… +op ๐‘‡))
 
Theoremhoadd32i 30506 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ((๐‘… +op ๐‘†) +op ๐‘‡) = ((๐‘… +op ๐‘‡) +op ๐‘†)
 
Theoremhocadddiri 30507 Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ((๐‘… +op ๐‘†) โˆ˜ ๐‘‡) = ((๐‘… โˆ˜ ๐‘‡) +op (๐‘† โˆ˜ ๐‘‡))
 
Theoremhocsubdiri 30508 Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ((๐‘… โˆ’op ๐‘†) โˆ˜ ๐‘‡) = ((๐‘… โˆ˜ ๐‘‡) โˆ’op (๐‘† โˆ˜ ๐‘‡))
 
Theoremho2coi 30509 Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (๐ด โˆˆ โ„‹ โ†’ (((๐‘… โˆ˜ ๐‘†) โˆ˜ ๐‘‡)โ€˜๐ด) = (๐‘…โ€˜(๐‘†โ€˜(๐‘‡โ€˜๐ด))))
 
Theoremhoaddass 30510 Associativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((๐‘…: โ„‹โŸถ โ„‹ โˆง ๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ ((๐‘… +op ๐‘†) +op ๐‘‡) = (๐‘… +op (๐‘† +op ๐‘‡)))
 
Theoremhoadd32 30511 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((๐‘…: โ„‹โŸถ โ„‹ โˆง ๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ ((๐‘… +op ๐‘†) +op ๐‘‡) = ((๐‘… +op ๐‘‡) +op ๐‘†))
 
Theoremhoadd4 30512 Rearrangement of 4 terms in a sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((๐‘…: โ„‹โŸถ โ„‹ โˆง ๐‘†: โ„‹โŸถ โ„‹) โˆง (๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹)) โ†’ ((๐‘… +op ๐‘†) +op (๐‘‡ +op ๐‘ˆ)) = ((๐‘… +op ๐‘‡) +op (๐‘† +op ๐‘ˆ)))
 
Theoremhocsubdir 30513 Distributive law for Hilbert space operator difference. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((๐‘…: โ„‹โŸถ โ„‹ โˆง ๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ ((๐‘… โˆ’op ๐‘†) โˆ˜ ๐‘‡) = ((๐‘… โˆ˜ ๐‘‡) โˆ’op (๐‘† โˆ˜ ๐‘‡)))
 
Theoremhoaddid1i 30514 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (๐‘‡ +op 0hop ) = ๐‘‡
 
Theoremhodidi 30515 Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (๐‘‡ โˆ’op ๐‘‡) = 0hop
 
Theoremho0coi 30516 Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ( 0hop โˆ˜ ๐‘‡) = 0hop
 
Theoremhoid1i 30517 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (๐‘‡ โˆ˜ Iop ) = ๐‘‡
 
Theoremhoid1ri 30518 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ( Iop โˆ˜ ๐‘‡) = ๐‘‡
 
Theoremhoaddid1 30519 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ (๐‘‡ +op 0hop ) = ๐‘‡)
 
Theoremhodid 30520 Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ (๐‘‡ โˆ’op ๐‘‡) = 0hop )
 
Theoremhon0 30521 A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ ยฌ ๐‘‡ = โˆ…)
 
Theoremhodseqi 30522 Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (๐‘† +op (๐‘‡ โˆ’op ๐‘†)) = ๐‘‡
 
Theoremho0subi 30523 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (๐‘† โˆ’op ๐‘‡) = (๐‘† +op ( 0hop โˆ’op ๐‘‡))
 
Theoremhonegsubi 30524 Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (๐‘† +op (-1 ยทop ๐‘‡)) = (๐‘† โˆ’op ๐‘‡)
 
Theoremho0sub 30525 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ (๐‘† โˆ’op ๐‘‡) = (๐‘† +op ( 0hop โˆ’op ๐‘‡)))
 
Theoremhosubid1 30526 The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ (๐‘‡ โˆ’op 0hop ) = ๐‘‡)
 
Theoremhonegsub 30527 Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (๐‘‡ +op (-1 ยทop ๐‘ˆ)) = (๐‘‡ โˆ’op ๐‘ˆ))
 
Theoremhomulid2 30528 An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ (1 ยทop ๐‘‡) = ๐‘‡)
 
Theoremhomco1 30529 Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ ((๐ด ยทop ๐‘‡) โˆ˜ ๐‘ˆ) = (๐ด ยทop (๐‘‡ โˆ˜ ๐‘ˆ)))
 
Theoremhomulass 30530 Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ ((๐ด ยท ๐ต) ยทop ๐‘‡) = (๐ด ยทop (๐ต ยทop ๐‘‡)))
 
Theoremhoadddi 30531 Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (๐ด ยทop (๐‘‡ +op ๐‘ˆ)) = ((๐ด ยทop ๐‘‡) +op (๐ด ยทop ๐‘ˆ)))
 
Theoremhoadddir 30532 Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ ((๐ด + ๐ต) ยทop ๐‘‡) = ((๐ด ยทop ๐‘‡) +op (๐ต ยทop ๐‘‡)))
 
Theoremhomul12 30533 Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ (๐ด ยทop (๐ต ยทop ๐‘‡)) = (๐ต ยทop (๐ด ยทop ๐‘‡)))
 
Theoremhonegneg 30534 Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ (-1 ยทop (-1 ยทop ๐‘‡)) = ๐‘‡)
 
Theoremhosubneg 30535 Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (๐‘‡ โˆ’op (-1 ยทop ๐‘ˆ)) = (๐‘‡ +op ๐‘ˆ))
 
Theoremhosubdi 30536 Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (๐ด ยทop (๐‘‡ โˆ’op ๐‘ˆ)) = ((๐ด ยทop ๐‘‡) โˆ’op (๐ด ยทop ๐‘ˆ)))
 
Theoremhonegdi 30537 Distribution of negative over addition. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (-1 ยทop (๐‘‡ +op ๐‘ˆ)) = ((-1 ยทop ๐‘‡) +op (-1 ยทop ๐‘ˆ)))
 
Theoremhonegsubdi 30538 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (-1 ยทop (๐‘‡ โˆ’op ๐‘ˆ)) = ((-1 ยทop ๐‘‡) +op ๐‘ˆ))
 
Theoremhonegsubdi2 30539 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (-1 ยทop (๐‘‡ โˆ’op ๐‘ˆ)) = (๐‘ˆ โˆ’op ๐‘‡))
 
Theoremhosubsub2 30540 Law for double subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (๐‘† โˆ’op (๐‘‡ โˆ’op ๐‘ˆ)) = (๐‘† +op (๐‘ˆ โˆ’op ๐‘‡)))
 
Theoremhosub4 30541 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((๐‘…: โ„‹โŸถ โ„‹ โˆง ๐‘†: โ„‹โŸถ โ„‹) โˆง (๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹)) โ†’ ((๐‘… +op ๐‘†) โˆ’op (๐‘‡ +op ๐‘ˆ)) = ((๐‘… โˆ’op ๐‘‡) +op (๐‘† โˆ’op ๐‘ˆ)))
 
Theoremhosubadd4 30542 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((๐‘…: โ„‹โŸถ โ„‹ โˆง ๐‘†: โ„‹โŸถ โ„‹) โˆง (๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹)) โ†’ ((๐‘… โˆ’op ๐‘†) โˆ’op (๐‘‡ โˆ’op ๐‘ˆ)) = ((๐‘… +op ๐‘ˆ) โˆ’op (๐‘† +op ๐‘‡)))
 
Theoremhoaddsubass 30543 Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ ((๐‘† +op ๐‘‡) โˆ’op ๐‘ˆ) = (๐‘† +op (๐‘‡ โˆ’op ๐‘ˆ)))
 
Theoremhoaddsub 30544 Law for operator addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ ((๐‘† +op ๐‘‡) โˆ’op ๐‘ˆ) = ((๐‘† โˆ’op ๐‘ˆ) +op ๐‘‡))
 
Theoremhosubsub 30545 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (๐‘† โˆ’op (๐‘‡ โˆ’op ๐‘ˆ)) = ((๐‘† โˆ’op ๐‘‡) +op ๐‘ˆ))
 
Theoremhosubsub4 30546 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹ โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ ((๐‘† โˆ’op ๐‘‡) โˆ’op ๐‘ˆ) = (๐‘† โˆ’op (๐‘‡ +op ๐‘ˆ)))
 
Theoremho2times 30547 Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ (2 ยทop ๐‘‡) = (๐‘‡ +op ๐‘‡))
 
Theoremhoaddsubassi 30548 Associativity of sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ((๐‘… +op ๐‘†) โˆ’op ๐‘‡) = (๐‘… +op (๐‘† โˆ’op ๐‘‡))
 
Theoremhoaddsubi 30549 Law for sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ((๐‘… +op ๐‘†) โˆ’op ๐‘‡) = ((๐‘… โˆ’op ๐‘‡) +op ๐‘†)
 
Theoremhosd1i 30550 Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    &   ๐‘ˆ: โ„‹โŸถ โ„‹    โ‡’   (๐‘‡ +op ๐‘ˆ) = (๐‘‡ โˆ’op ( 0hop โˆ’op ๐‘ˆ))
 
Theoremhosd2i 30551 Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    &   ๐‘ˆ: โ„‹โŸถ โ„‹    โ‡’   (๐‘‡ +op ๐‘ˆ) = (๐‘‡ โˆ’op ((๐‘ˆ โˆ’op ๐‘ˆ) โˆ’op ๐‘ˆ))
 
Theoremhopncani 30552 Hilbert space operator cancellation law. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    &   ๐‘ˆ: โ„‹โŸถ โ„‹    โ‡’   ((๐‘‡ +op ๐‘ˆ) โˆ’op ๐‘ˆ) = ๐‘‡
 
Theoremhonpcani 30553 Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    &   ๐‘ˆ: โ„‹โŸถ โ„‹    โ‡’   ((๐‘‡ โˆ’op ๐‘ˆ) +op ๐‘ˆ) = ๐‘‡
 
Theoremhosubeq0i 30554 If the difference between two operators is zero, they are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    &   ๐‘ˆ: โ„‹โŸถ โ„‹    โ‡’   ((๐‘‡ โˆ’op ๐‘ˆ) = 0hop โ†” ๐‘‡ = ๐‘ˆ)
 
Theoremhonpncani 30555 Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
๐‘…: โ„‹โŸถ โ„‹    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   ((๐‘… โˆ’op ๐‘†) +op (๐‘† โˆ’op ๐‘‡)) = (๐‘… โˆ’op ๐‘‡)
 
Theoremho01i 30556* A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S8) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ ((๐‘‡โ€˜๐‘ฅ) ยทih ๐‘ฆ) = 0 โ†” ๐‘‡ = 0hop )
 
Theoremho02i 30557* A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S10) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฆ)) = 0 โ†” ๐‘‡ = 0hop )
 
Theoremhoeq1 30558* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ (โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ ((๐‘†โ€˜๐‘ฅ) ยทih ๐‘ฆ) = ((๐‘‡โ€˜๐‘ฅ) ยทih ๐‘ฆ) โ†” ๐‘† = ๐‘‡))
 
Theoremhoeq2 30559* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ (โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘†โ€˜๐‘ฆ)) = (๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฆ)) โ†” ๐‘† = ๐‘‡))
 
Theoremadjmo 30560* Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
โˆƒ*๐‘ข(๐‘ข: โ„‹โŸถ โ„‹ โˆง โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฆ)) = ((๐‘ขโ€˜๐‘ฅ) ยทih ๐‘ฆ))
 
Theoremadjsym 30561* Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
((๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ (โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘†โ€˜๐‘ฆ)) = ((๐‘‡โ€˜๐‘ฅ) ยทih ๐‘ฆ) โ†” โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฆ)) = ((๐‘†โ€˜๐‘ฅ) ยทih ๐‘ฆ)))
 
Theoremeigrei 30562 A necessary and sufficient condition (that holds when ๐‘‡ is a Hermitian operator) for an eigenvalue ๐ต to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 21-Jan-2005.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‚    โ‡’   (((๐‘‡โ€˜๐ด) = (๐ต ยทโ„Ž ๐ด) โˆง ๐ด โ‰  0โ„Ž) โ†’ ((๐ด ยทih (๐‘‡โ€˜๐ด)) = ((๐‘‡โ€˜๐ด) ยทih ๐ด) โ†” ๐ต โˆˆ โ„))
 
Theoremeigre 30563 A necessary and sufficient condition (that holds when ๐‘‡ is a Hermitian operator) for an eigenvalue ๐ต to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
(((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‚) โˆง ((๐‘‡โ€˜๐ด) = (๐ต ยทโ„Ž ๐ด) โˆง ๐ด โ‰  0โ„Ž)) โ†’ ((๐ด ยทih (๐‘‡โ€˜๐ด)) = ((๐‘‡โ€˜๐ด) ยทih ๐ด) โ†” ๐ต โˆˆ โ„))
 
Theoremeigposi 30564 A sufficient condition (first conjunct pair, that holds when ๐‘‡ is a positive operator) for an eigenvalue ๐ต (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‚    โ‡’   ((((๐ด ยทih (๐‘‡โ€˜๐ด)) โˆˆ โ„ โˆง 0 โ‰ค (๐ด ยทih (๐‘‡โ€˜๐ด))) โˆง ((๐‘‡โ€˜๐ด) = (๐ต ยทโ„Ž ๐ด) โˆง ๐ด โ‰  0โ„Ž)) โ†’ (๐ต โˆˆ โ„ โˆง 0 โ‰ค ๐ต))
 
Theoremeigorthi 30565 A necessary and sufficient condition (that holds when ๐‘‡ is a Hermitian operator) for two eigenvectors ๐ด and ๐ต to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‚    &   ๐ท โˆˆ โ„‚    โ‡’   ((((๐‘‡โ€˜๐ด) = (๐ถ ยทโ„Ž ๐ด) โˆง (๐‘‡โ€˜๐ต) = (๐ท ยทโ„Ž ๐ต)) โˆง ๐ถ โ‰  (โˆ—โ€˜๐ท)) โ†’ ((๐ด ยทih (๐‘‡โ€˜๐ต)) = ((๐‘‡โ€˜๐ด) ยทih ๐ต) โ†” (๐ด ยทih ๐ต) = 0))
 
Theoremeigorth 30566 A necessary and sufficient condition (that holds when ๐‘‡ is a Hermitian operator) for two eigenvectors ๐ด and ๐ต to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
((((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โˆง (๐ถ โˆˆ โ„‚ โˆง ๐ท โˆˆ โ„‚)) โˆง (((๐‘‡โ€˜๐ด) = (๐ถ ยทโ„Ž ๐ด) โˆง (๐‘‡โ€˜๐ต) = (๐ท ยทโ„Ž ๐ต)) โˆง ๐ถ โ‰  (โˆ—โ€˜๐ท))) โ†’ ((๐ด ยทih (๐‘‡โ€˜๐ต)) = ((๐‘‡โ€˜๐ด) ยทih ๐ต) โ†” (๐ด ยทih ๐ต) = 0))
 
20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms
 
Definitiondf-nmop 30567* Define the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
normop = (๐‘ก โˆˆ ( โ„‹ โ†‘m โ„‹) โ†ฆ sup({๐‘ฅ โˆฃ โˆƒ๐‘ง โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ง) โ‰ค 1 โˆง ๐‘ฅ = (normโ„Žโ€˜(๐‘กโ€˜๐‘ง)))}, โ„*, < ))
 
Definitiondf-cnop 30568* Define the set of continuous operators on Hilbert space. For every "epsilon" (๐‘ฆ) there is a "delta" (๐‘ง) such that... (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
ContOp = {๐‘ก โˆˆ ( โ„‹ โ†‘m โ„‹) โˆฃ โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„+ โˆƒ๐‘ง โˆˆ โ„+ โˆ€๐‘ค โˆˆ โ„‹ ((normโ„Žโ€˜(๐‘ค โˆ’โ„Ž ๐‘ฅ)) < ๐‘ง โ†’ (normโ„Žโ€˜((๐‘กโ€˜๐‘ค) โˆ’โ„Ž (๐‘กโ€˜๐‘ฅ))) < ๐‘ฆ)}
 
Definitiondf-lnop 30569* Define the set of linear operators on Hilbert space. (See df-hosum 30458 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
LinOp = {๐‘ก โˆˆ ( โ„‹ โ†‘m โ„‹) โˆฃ โˆ€๐‘ฅ โˆˆ โ„‚ โˆ€๐‘ฆ โˆˆ โ„‹ โˆ€๐‘ง โˆˆ โ„‹ (๐‘กโ€˜((๐‘ฅ ยทโ„Ž ๐‘ฆ) +โ„Ž ๐‘ง)) = ((๐‘ฅ ยทโ„Ž (๐‘กโ€˜๐‘ฆ)) +โ„Ž (๐‘กโ€˜๐‘ง))}
 
Definitiondf-bdop 30570 Define the set of bounded linear Hilbert space operators. (See df-hosum 30458 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
BndLinOp = {๐‘ก โˆˆ LinOp โˆฃ (normopโ€˜๐‘ก) < +โˆž}
 
Definitiondf-unop 30571* Define the set of unitary operators on Hilbert space. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
UniOp = {๐‘ก โˆฃ (๐‘ก: โ„‹โ€“ontoโ†’ โ„‹ โˆง โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ ((๐‘กโ€˜๐‘ฅ) ยทih (๐‘กโ€˜๐‘ฆ)) = (๐‘ฅ ยทih ๐‘ฆ))}
 
Definitiondf-hmop 30572* Define the set of Hermitian operators on Hilbert space. Some books call these "symmetric operators" and others call them "self-adjoint operators", sometimes with slightly different technical meanings. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
HrmOp = {๐‘ก โˆˆ ( โ„‹ โ†‘m โ„‹) โˆฃ โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘กโ€˜๐‘ฆ)) = ((๐‘กโ€˜๐‘ฅ) ยทih ๐‘ฆ)}
 
20.6.5  Linear and continuous functionals and norms
 
Definitiondf-nmfn 30573* Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
normfn = (๐‘ก โˆˆ (โ„‚ โ†‘m โ„‹) โ†ฆ sup({๐‘ฅ โˆฃ โˆƒ๐‘ง โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ง) โ‰ค 1 โˆง ๐‘ฅ = (absโ€˜(๐‘กโ€˜๐‘ง)))}, โ„*, < ))
 
Definitiondf-nlfn 30574 Define the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
null = (๐‘ก โˆˆ (โ„‚ โ†‘m โ„‹) โ†ฆ (โ—ก๐‘ก โ€œ {0}))
 
Definitiondf-cnfn 30575* Define the set of continuous functionals on Hilbert space. For every "epsilon" (๐‘ฆ) there is a "delta" (๐‘ง) such that... (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
ContFn = {๐‘ก โˆˆ (โ„‚ โ†‘m โ„‹) โˆฃ โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„+ โˆƒ๐‘ง โˆˆ โ„+ โˆ€๐‘ค โˆˆ โ„‹ ((normโ„Žโ€˜(๐‘ค โˆ’โ„Ž ๐‘ฅ)) < ๐‘ง โ†’ (absโ€˜((๐‘กโ€˜๐‘ค) โˆ’ (๐‘กโ€˜๐‘ฅ))) < ๐‘ฆ)}
 
Definitiondf-lnfn 30576* Define the set of linear functionals on Hilbert space. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
LinFn = {๐‘ก โˆˆ (โ„‚ โ†‘m โ„‹) โˆฃ โˆ€๐‘ฅ โˆˆ โ„‚ โˆ€๐‘ฆ โˆˆ โ„‹ โˆ€๐‘ง โˆˆ โ„‹ (๐‘กโ€˜((๐‘ฅ ยทโ„Ž ๐‘ฆ) +โ„Ž ๐‘ง)) = ((๐‘ฅ ยท (๐‘กโ€˜๐‘ฆ)) + (๐‘กโ€˜๐‘ง))}
 
20.6.6  Adjoint
 
Definitiondf-adjh 30577* Define the adjoint of a Hilbert space operator (if it exists). The domain of adjโ„Ž is the set of all adjoint operators. Definition of adjoint in [Kalmbach2] p. 8. Unlike Kalmbach (and most authors), we do not demand that the operator be linear, but instead show (in adjbdln 30811) that the adjoint exists for a bounded linear operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
adjโ„Ž = {โŸจ๐‘ก, ๐‘ขโŸฉ โˆฃ (๐‘ก: โ„‹โŸถ โ„‹ โˆง ๐‘ข: โ„‹โŸถ โ„‹ โˆง โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ ((๐‘กโ€˜๐‘ฅ) ยทih ๐‘ฆ) = (๐‘ฅ ยทih (๐‘ขโ€˜๐‘ฆ)))}
 
20.6.7  Dirac bra-ket notation
 
Definitiondf-bra 30578* Define the bra of a vector used by Dirac notation. Based on definition of bra in [Prugovecki] p. 186 (p. 180 in 1971 edition). In Dirac bra-ket notation, โŸจ๐ด โˆฃ ๐ตโŸฉ is a complex number equal to the inner product (๐ต ยทih ๐ด). But physicists like to talk about the individual components โŸจ๐ด โˆฃ and โˆฃ ๐ตโŸฉ, called bra and ket respectively. In order for their properties to make sense formally, we define the ket โˆฃ ๐ตโŸฉ as the vector ๐ต itself, and the bra โŸจ๐ด โˆฃ as a functional from โ„‹ to โ„‚. We represent the Dirac notation โŸจ๐ด โˆฃ ๐ตโŸฉ by ((braโ€˜๐ด)โ€˜๐ต); see braval 30672. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 29812) but is also required in order for the associative law kbass2 30845 to work.

Our definition of bra and the associated outer product df-kb 30579 differs from, but is equivalent to, a common approach in the literature that makes use of mappings to a dual space. Our approach eliminates the need to have a parallel development of this dual space and instead keeps everything in Hilbert space.

For an extensive discussion about how our notation maps to the bra-ket notation in physics textbooks, see mmnotes.txt 30579, under the 17-May-2006 entry. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)

bra = (๐‘ฅ โˆˆ โ„‹ โ†ฆ (๐‘ฆ โˆˆ โ„‹ โ†ฆ (๐‘ฆ ยทih ๐‘ฅ)))
 
Definitiondf-kb 30579* Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation, โˆฃ ๐ดโŸฉโŸจ๐ต โˆฃ is an operator known as the outer product of ๐ด and ๐ต, which we represent by (๐ด ketbra ๐ต). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with Definition df-bra 30578, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
ketbra = (๐‘ฅ โˆˆ โ„‹, ๐‘ฆ โˆˆ โ„‹ โ†ฆ (๐‘ง โˆˆ โ„‹ โ†ฆ ((๐‘ง ยทih ๐‘ฆ) ยทโ„Ž ๐‘ฅ)))
 
20.6.8  Positive operators
 
Definitiondf-leop 30580* Define positive operator ordering. Definition VI.1 of [Retherford] p. 49. Note that ( โ„‹ ร— 0โ„‹) โ‰คop ๐‘‡ means that ๐‘‡ is a positive operator. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
โ‰คop = {โŸจ๐‘ก, ๐‘ขโŸฉ โˆฃ ((๐‘ข โˆ’op ๐‘ก) โˆˆ HrmOp โˆง โˆ€๐‘ฅ โˆˆ โ„‹ 0 โ‰ค (((๐‘ข โˆ’op ๐‘ก)โ€˜๐‘ฅ) ยทih ๐‘ฅ))}
 
20.6.9  Eigenvectors, eigenvalues, spectrum
 
Definitiondf-eigvec 30581* Define the eigenvector function. Theorem eleigveccl 30687 shows that eigvecโ€˜๐‘‡, the set of eigenvectors of Hilbert space operator ๐‘‡, are Hilbert space vectors. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
eigvec = (๐‘ก โˆˆ ( โ„‹ โ†‘m โ„‹) โ†ฆ {๐‘ฅ โˆˆ ( โ„‹ โˆ– 0โ„‹) โˆฃ โˆƒ๐‘ง โˆˆ โ„‚ (๐‘กโ€˜๐‘ฅ) = (๐‘ง ยทโ„Ž ๐‘ฅ)})
 
Definitiondf-eigval 30582* Define the eigenvalue function. The range of eigvalโ€˜๐‘‡ is the set of eigenvalues of Hilbert space operator ๐‘‡. Theorem eigvalcl 30689 shows that (eigvalโ€˜๐‘‡)โ€˜๐ด, the eigenvalue associated with eigenvector ๐ด, is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
eigval = (๐‘ก โˆˆ ( โ„‹ โ†‘m โ„‹) โ†ฆ (๐‘ฅ โˆˆ (eigvecโ€˜๐‘ก) โ†ฆ (((๐‘กโ€˜๐‘ฅ) ยทih ๐‘ฅ) / ((normโ„Žโ€˜๐‘ฅ)โ†‘2))))
 
Definitiondf-spec 30583* Define the spectrum of an operator. Definition of spectrum in [Halmos] p. 50. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
Lambda = (๐‘ก โˆˆ ( โ„‹ โ†‘m โ„‹) โ†ฆ {๐‘ฅ โˆˆ โ„‚ โˆฃ ยฌ (๐‘ก โˆ’op (๐‘ฅ ยทop ( I โ†พ โ„‹))): โ„‹โ€“1-1โ†’ โ„‹})
 
20.6.10  Theorems about operators and functionals
 
Theoremnmopval 30584* Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ (normopโ€˜๐‘‡) = sup({๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ฆ) โ‰ค 1 โˆง ๐‘ฅ = (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฆ)))}, โ„*, < ))
 
Theoremelcnop 30585* Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(๐‘‡ โˆˆ ContOp โ†” (๐‘‡: โ„‹โŸถ โ„‹ โˆง โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„+ โˆƒ๐‘ง โˆˆ โ„+ โˆ€๐‘ค โˆˆ โ„‹ ((normโ„Žโ€˜(๐‘ค โˆ’โ„Ž ๐‘ฅ)) < ๐‘ง โ†’ (normโ„Žโ€˜((๐‘‡โ€˜๐‘ค) โˆ’โ„Ž (๐‘‡โ€˜๐‘ฅ))) < ๐‘ฆ)))
 
Theoremellnop 30586* Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinOp โ†” (๐‘‡: โ„‹โŸถ โ„‹ โˆง โˆ€๐‘ฅ โˆˆ โ„‚ โˆ€๐‘ฆ โˆˆ โ„‹ โˆ€๐‘ง โˆˆ โ„‹ (๐‘‡โ€˜((๐‘ฅ ยทโ„Ž ๐‘ฆ) +โ„Ž ๐‘ง)) = ((๐‘ฅ ยทโ„Ž (๐‘‡โ€˜๐‘ฆ)) +โ„Ž (๐‘‡โ€˜๐‘ง))))
 
Theoremlnopf 30587 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinOp โ†’ ๐‘‡: โ„‹โŸถ โ„‹)
 
Theoremelbdop 30588 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(๐‘‡ โˆˆ BndLinOp โ†” (๐‘‡ โˆˆ LinOp โˆง (normopโ€˜๐‘‡) < +โˆž))
 
Theorembdopln 30589 A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ BndLinOp โ†’ ๐‘‡ โˆˆ LinOp)
 
Theorembdopf 30590 A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ BndLinOp โ†’ ๐‘‡: โ„‹โŸถ โ„‹)
 
TheoremnmopsetretALT 30591* The set in the supremum of the operator norm definition df-nmop 30567 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ฆ) โ‰ค 1 โˆง ๐‘ฅ = (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฆ)))} โŠ† โ„)
 
TheoremnmopsetretHIL 30592* The set in the supremum of the operator norm definition df-nmop 30567 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ฆ) โ‰ค 1 โˆง ๐‘ฅ = (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฆ)))} โŠ† โ„)
 
Theoremnmopsetn0 30593* The set in the supremum of the operator norm definition df-nmop 30567 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(normโ„Žโ€˜(๐‘‡โ€˜0โ„Ž)) โˆˆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ฆ) โ‰ค 1 โˆง ๐‘ฅ = (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฆ)))}
 
Theoremnmopxr 30594 The norm of a Hilbert space operator is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ (normopโ€˜๐‘‡) โˆˆ โ„*)
 
Theoremnmoprepnf 30595 The norm of a Hilbert space operator is either real or plus infinity. (Contributed by NM, 5-Feb-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ ((normopโ€˜๐‘‡) โˆˆ โ„ โ†” (normopโ€˜๐‘‡) โ‰  +โˆž))
 
Theoremnmopgtmnf 30596 The norm of a Hilbert space operator is not minus infinity. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ -โˆž < (normopโ€˜๐‘‡))
 
Theoremnmopreltpnf 30597 The norm of a Hilbert space operator is real iff it is less than infinity. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถ โ„‹ โ†’ ((normopโ€˜๐‘‡) โˆˆ โ„ โ†” (normopโ€˜๐‘‡) < +โˆž))
 
Theoremnmopre 30598 The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ BndLinOp โ†’ (normopโ€˜๐‘‡) โˆˆ โ„)
 
Theoremelbdop2 30599 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ BndLinOp โ†” (๐‘‡ โˆˆ LinOp โˆง (normopโ€˜๐‘‡) โˆˆ โ„))
 
Theoremelunop 30600* Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ UniOp โ†” (๐‘‡: โ„‹โ€“ontoโ†’ โ„‹ โˆง โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ ((๐‘‡โ€˜๐‘ฅ) ยทih (๐‘‡โ€˜๐‘ฆ)) = (๐‘ฅ ยทih ๐‘ฆ)))
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