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Theorem List for Metamath Proof Explorer - 31301-31400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremho2coi 31301 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 31302 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 31303 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 31304 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 31305 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‘‡)))
 
Theoremhoaddridi 31306 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 31307 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 31308 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 31309 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 31310 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‘‡
 
Theoremhoaddrid 31311 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 31312 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 31313 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 31314 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 31315 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 31316 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 31317 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 31318 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 31319 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))
 
Theoremhomullid 31320 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 31321 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 31322 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 31323 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 31324 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 31325 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 31326 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 31327 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 31328 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 31329 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 31330 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 31331 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 31332 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 31333 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 31334 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 31335 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 31336 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 31337 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 31338 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 31339 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 31340 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 31341 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 31342 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 31343 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 31344 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 31345 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 31346 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 31347 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 31348* 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 31349* 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 31350* 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 31351* 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 31352* 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 31353* 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 31354 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 31355 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 31356 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 31357 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 31358 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 31359* 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 31360* 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 31361* Define the set of linear operators on Hilbert space. (See df-hosum 31250 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 31362 Define the set of bounded linear Hilbert space operators. (See df-hosum 31250 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 31363* 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 31364* 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 31365* 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 31366 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 31367* 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 31368* 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 31369* 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 31603) 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 31370* 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 31464. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 30604) but is also required in order for the associative law kbass2 31637 to work.

Our definition of bra and the associated outer product df-kb 31371 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 31371, 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 31371* 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 31370, 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 31372* 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 31373* Define the eigenvector function. Theorem eleigveccl 31479 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 31374* Define the eigenvalue function. The range of eigvalโ€˜๐‘‡ is the set of eigenvalues of Hilbert space operator ๐‘‡. Theorem eigvalcl 31481 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 31375* 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 31376* 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 31377* 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 31378* 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 31379 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 31380 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 31381 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 31382 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 31383* The set in the supremum of the operator norm definition df-nmop 31359 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 31384* The set in the supremum of the operator norm definition df-nmop 31359 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 31385* The set in the supremum of the operator norm definition df-nmop 31359 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 31386 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 31387 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 31388 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 31389 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 31390 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 31391 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 31392* 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)))
 
Theoremelhmop 31393* Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(๐‘‡ โˆˆ HrmOp โ†” (๐‘‡: โ„‹โŸถ โ„‹ โˆง โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ฆ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฆ)) = ((๐‘‡โ€˜๐‘ฅ) ยทih ๐‘ฆ)))
 
Theoremhmopf 31394 A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ HrmOp โ†’ ๐‘‡: โ„‹โŸถ โ„‹)
 
Theoremhmopex 31395 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
HrmOp โˆˆ V
 
Theoremnmfnval 31396* Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถโ„‚ โ†’ (normfnโ€˜๐‘‡) = sup({๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ฆ) โ‰ค 1 โˆง ๐‘ฅ = (absโ€˜(๐‘‡โ€˜๐‘ฆ)))}, โ„*, < ))
 
Theoremnmfnsetre 31397* The set in the supremum of the functional norm definition df-nmfn 31365 is a set of reals. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถโ„‚ โ†’ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ฆ) โ‰ค 1 โˆง ๐‘ฅ = (absโ€˜(๐‘‡โ€˜๐‘ฆ)))} โŠ† โ„)
 
Theoremnmfnsetn0 31398* The set in the supremum of the functional norm definition df-nmfn 31365 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(absโ€˜(๐‘‡โ€˜0โ„Ž)) โˆˆ {๐‘ฅ โˆฃ โˆƒ๐‘ฆ โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ฆ) โ‰ค 1 โˆง ๐‘ฅ = (absโ€˜(๐‘‡โ€˜๐‘ฆ)))}
 
Theoremnmfnxr 31399 The norm of any Hilbert space functional is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถโ„‚ โ†’ (normfnโ€˜๐‘‡) โˆˆ โ„*)
 
Theoremnmfnrepnf 31400 The norm of a Hilbert space functional is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
(๐‘‡: โ„‹โŸถโ„‚ โ†’ ((normfnโ€˜๐‘‡) โˆˆ โ„ โ†” (normfnโ€˜๐‘‡) โ‰  +โˆž))
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