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Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhicli 30601 Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   (๐ด ยทih ๐ต) โˆˆ โ„‚
 
Axiomax-his1 30602 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that โˆ—โ€˜๐‘ฅ is the complex conjugate cjval 15053 of ๐‘ฅ. In the literature, the inner product of ๐ด and ๐ต is usually written โŸจ๐ด, ๐ตโŸฉ, but our operation notation co 7411 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4634. Physicists use โŸจ๐ต โˆฃ ๐ดโŸฉ, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 31370. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (๐ด ยทih ๐ต) = (โˆ—โ€˜(๐ต ยทih ๐ด)))
 
Axiomax-his2 30603 Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ ((๐ด +โ„Ž ๐ต) ยทih ๐ถ) = ((๐ด ยทih ๐ถ) + (๐ต ยทih ๐ถ)))
 
Axiomax-his3 30604 Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with (๐ต ยทih (๐ด ยทโ„Ž ๐ถ)) (e.g., Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 31370 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ ((๐ด ยทโ„Ž ๐ต) ยทih ๐ถ) = (๐ด ยท (๐ต ยทih ๐ถ)))
 
Axiomax-his4 30605 Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ด โ‰  0โ„Ž) โ†’ 0 < (๐ด ยทih ๐ด))
 
20.2  Inner product and norms
 
20.2.1  Inner product
 
Theoremhis5 30606 Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (๐ต ยทih (๐ด ยทโ„Ž ๐ถ)) = ((โˆ—โ€˜๐ด) ยท (๐ต ยทih ๐ถ)))
 
Theoremhis52 30607 Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (๐ต ยทih ((โˆ—โ€˜๐ด) ยทโ„Ž ๐ถ)) = (๐ด ยท (๐ต ยทih ๐ถ)))
 
Theoremhis35 30608 Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚) โˆง (๐ถ โˆˆ โ„‹ โˆง ๐ท โˆˆ โ„‹)) โ†’ ((๐ด ยทโ„Ž ๐ถ) ยทih (๐ต ยทโ„Ž ๐ท)) = ((๐ด ยท (โˆ—โ€˜๐ต)) ยท (๐ถ ยทih ๐ท)))
 
Theoremhis35i 30609 Move scalar multiplication to outside of inner product. (Contributed by NM, 1-Jul-2005.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‚    &   ๐ถ โˆˆ โ„‹    &   ๐ท โˆˆ โ„‹    โ‡’   ((๐ด ยทโ„Ž ๐ถ) ยทih (๐ต ยทโ„Ž ๐ท)) = ((๐ด ยท (โˆ—โ€˜๐ต)) ยท (๐ถ ยทih ๐ท))
 
Theoremhis7 30610 Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (๐ด ยทih (๐ต +โ„Ž ๐ถ)) = ((๐ด ยทih ๐ต) + (๐ด ยทih ๐ถ)))
 
Theoremhiassdi 30611 Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
(((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹) โˆง (๐ถ โˆˆ โ„‹ โˆง ๐ท โˆˆ โ„‹)) โ†’ (((๐ด ยทโ„Ž ๐ต) +โ„Ž ๐ถ) ยทih ๐ท) = ((๐ด ยท (๐ต ยทih ๐ท)) + (๐ถ ยทih ๐ท)))
 
Theoremhis2sub 30612 Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ ((๐ด โˆ’โ„Ž ๐ต) ยทih ๐ถ) = ((๐ด ยทih ๐ถ) โˆ’ (๐ต ยทih ๐ถ)))
 
Theoremhis2sub2 30613 Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (๐ด ยทih (๐ต โˆ’โ„Ž ๐ถ)) = ((๐ด ยทih ๐ต) โˆ’ (๐ด ยทih ๐ถ)))
 
Theoremhire 30614 A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ ((๐ด ยทih ๐ต) โˆˆ โ„ โ†” (๐ด ยทih ๐ต) = (๐ต ยทih ๐ด)))
 
Theoremhiidrcl 30615 Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ (๐ด ยทih ๐ด) โˆˆ โ„)
 
Theoremhi01 30616 Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ (0โ„Ž ยทih ๐ด) = 0)
 
Theoremhi02 30617 Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ (๐ด ยทih 0โ„Ž) = 0)
 
Theoremhiidge0 30618 Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ 0 โ‰ค (๐ด ยทih ๐ด))
 
Theoremhis6 30619 Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ ((๐ด ยทih ๐ด) = 0 โ†” ๐ด = 0โ„Ž))
 
Theoremhis1i 30620 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   (๐ด ยทih ๐ต) = (โˆ—โ€˜(๐ต ยทih ๐ด))
 
Theoremabshicom 30621 Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (absโ€˜(๐ด ยทih ๐ต)) = (absโ€˜(๐ต ยทih ๐ด)))
 
Theoremhial0 30622* A vector whose inner product is always zero is zero. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ (โˆ€๐‘ฅ โˆˆ โ„‹ (๐ด ยทih ๐‘ฅ) = 0 โ†” ๐ด = 0โ„Ž))
 
Theoremhial02 30623* A vector whose inner product is always zero is zero. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ (โˆ€๐‘ฅ โˆˆ โ„‹ (๐‘ฅ ยทih ๐ด) = 0 โ†” ๐ด = 0โ„Ž))
 
Theoremhisubcomi 30624 Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‹    &   ๐ท โˆˆ โ„‹    โ‡’   ((๐ด โˆ’โ„Ž ๐ต) ยทih (๐ถ โˆ’โ„Ž ๐ท)) = ((๐ต โˆ’โ„Ž ๐ด) ยทih (๐ท โˆ’โ„Ž ๐ถ))
 
Theoremhi2eq 30625 Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ ((๐ด ยทih (๐ด โˆ’โ„Ž ๐ต)) = (๐ต ยทih (๐ด โˆ’โ„Ž ๐ต)) โ†” ๐ด = ๐ต))
 
Theoremhial2eq 30626* Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (โˆ€๐‘ฅ โˆˆ โ„‹ (๐ด ยทih ๐‘ฅ) = (๐ต ยทih ๐‘ฅ) โ†” ๐ด = ๐ต))
 
Theoremhial2eq2 30627* Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (โˆ€๐‘ฅ โˆˆ โ„‹ (๐‘ฅ ยทih ๐ด) = (๐‘ฅ ยทih ๐ต) โ†” ๐ด = ๐ต))
 
Theoremorthcom 30628 Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ ((๐ด ยทih ๐ต) = 0 โ†” (๐ต ยทih ๐ด) = 0))
 
Theoremnormlem0 30629 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 7-Oct-1999.) (New usage is discouraged.)
๐‘† โˆˆ โ„‚    &   ๐น โˆˆ โ„‹    &   ๐บ โˆˆ โ„‹    โ‡’   ((๐น โˆ’โ„Ž (๐‘† ยทโ„Ž ๐บ)) ยทih (๐น โˆ’โ„Ž (๐‘† ยทโ„Ž ๐บ))) = (((๐น ยทih ๐น) + (-(โˆ—โ€˜๐‘†) ยท (๐น ยทih ๐บ))) + ((-๐‘† ยท (๐บ ยทih ๐น)) + ((๐‘† ยท (โˆ—โ€˜๐‘†)) ยท (๐บ ยทih ๐บ))))
 
Theoremnormlem1 30630 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 22-Aug-1999.) (New usage is discouraged.)
๐‘† โˆˆ โ„‚    &   ๐น โˆˆ โ„‹    &   ๐บ โˆˆ โ„‹    &   ๐‘… โˆˆ โ„    &   (absโ€˜๐‘†) = 1    โ‡’   ((๐น โˆ’โ„Ž ((๐‘† ยท ๐‘…) ยทโ„Ž ๐บ)) ยทih (๐น โˆ’โ„Ž ((๐‘† ยท ๐‘…) ยทโ„Ž ๐บ))) = (((๐น ยทih ๐น) + (((โˆ—โ€˜๐‘†) ยท -๐‘…) ยท (๐น ยทih ๐บ))) + (((๐‘† ยท -๐‘…) ยท (๐บ ยทih ๐น)) + ((๐‘…โ†‘2) ยท (๐บ ยทih ๐บ))))
 
Theoremnormlem2 30631 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
๐‘† โˆˆ โ„‚    &   ๐น โˆˆ โ„‹    &   ๐บ โˆˆ โ„‹    &   ๐ต = -(((โˆ—โ€˜๐‘†) ยท (๐น ยทih ๐บ)) + (๐‘† ยท (๐บ ยทih ๐น)))    โ‡’   ๐ต โˆˆ โ„
 
Theoremnormlem3 30632 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
๐‘† โˆˆ โ„‚    &   ๐น โˆˆ โ„‹    &   ๐บ โˆˆ โ„‹    &   ๐ต = -(((โˆ—โ€˜๐‘†) ยท (๐น ยทih ๐บ)) + (๐‘† ยท (๐บ ยทih ๐น)))    &   ๐ด = (๐บ ยทih ๐บ)    &   ๐ถ = (๐น ยทih ๐น)    &   ๐‘… โˆˆ โ„    โ‡’   (((๐ด ยท (๐‘…โ†‘2)) + (๐ต ยท ๐‘…)) + ๐ถ) = (((๐น ยทih ๐น) + (((โˆ—โ€˜๐‘†) ยท -๐‘…) ยท (๐น ยทih ๐บ))) + (((๐‘† ยท -๐‘…) ยท (๐บ ยทih ๐น)) + ((๐‘…โ†‘2) ยท (๐บ ยทih ๐บ))))
 
Theoremnormlem4 30633 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
๐‘† โˆˆ โ„‚    &   ๐น โˆˆ โ„‹    &   ๐บ โˆˆ โ„‹    &   ๐ต = -(((โˆ—โ€˜๐‘†) ยท (๐น ยทih ๐บ)) + (๐‘† ยท (๐บ ยทih ๐น)))    &   ๐ด = (๐บ ยทih ๐บ)    &   ๐ถ = (๐น ยทih ๐น)    &   ๐‘… โˆˆ โ„    &   (absโ€˜๐‘†) = 1    โ‡’   ((๐น โˆ’โ„Ž ((๐‘† ยท ๐‘…) ยทโ„Ž ๐บ)) ยทih (๐น โˆ’โ„Ž ((๐‘† ยท ๐‘…) ยทโ„Ž ๐บ))) = (((๐ด ยท (๐‘…โ†‘2)) + (๐ต ยท ๐‘…)) + ๐ถ)
 
Theoremnormlem5 30634 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Aug-1999.) (New usage is discouraged.)
๐‘† โˆˆ โ„‚    &   ๐น โˆˆ โ„‹    &   ๐บ โˆˆ โ„‹    &   ๐ต = -(((โˆ—โ€˜๐‘†) ยท (๐น ยทih ๐บ)) + (๐‘† ยท (๐บ ยทih ๐น)))    &   ๐ด = (๐บ ยทih ๐บ)    &   ๐ถ = (๐น ยทih ๐น)    &   ๐‘… โˆˆ โ„    &   (absโ€˜๐‘†) = 1    โ‡’   0 โ‰ค (((๐ด ยท (๐‘…โ†‘2)) + (๐ต ยท ๐‘…)) + ๐ถ)
 
Theoremnormlem6 30635 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) (New usage is discouraged.)
๐‘† โˆˆ โ„‚    &   ๐น โˆˆ โ„‹    &   ๐บ โˆˆ โ„‹    &   ๐ต = -(((โˆ—โ€˜๐‘†) ยท (๐น ยทih ๐บ)) + (๐‘† ยท (๐บ ยทih ๐น)))    &   ๐ด = (๐บ ยทih ๐บ)    &   ๐ถ = (๐น ยทih ๐น)    &   (absโ€˜๐‘†) = 1    โ‡’   (absโ€˜๐ต) โ‰ค (2 ยท ((โˆšโ€˜๐ด) ยท (โˆšโ€˜๐ถ)))
 
Theoremnormlem7 30636 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
๐‘† โˆˆ โ„‚    &   ๐น โˆˆ โ„‹    &   ๐บ โˆˆ โ„‹    &   (absโ€˜๐‘†) = 1    โ‡’   (((โˆ—โ€˜๐‘†) ยท (๐น ยทih ๐บ)) + (๐‘† ยท (๐บ ยทih ๐น))) โ‰ค (2 ยท ((โˆšโ€˜(๐บ ยทih ๐บ)) ยท (โˆšโ€˜(๐น ยทih ๐น))))
 
Theoremnormlem8 30637 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‹    &   ๐ท โˆˆ โ„‹    โ‡’   ((๐ด +โ„Ž ๐ต) ยทih (๐ถ +โ„Ž ๐ท)) = (((๐ด ยทih ๐ถ) + (๐ต ยทih ๐ท)) + ((๐ด ยทih ๐ท) + (๐ต ยทih ๐ถ)))
 
Theoremnormlem9 30638 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‹    &   ๐ท โˆˆ โ„‹    โ‡’   ((๐ด โˆ’โ„Ž ๐ต) ยทih (๐ถ โˆ’โ„Ž ๐ท)) = (((๐ด ยทih ๐ถ) + (๐ต ยทih ๐ท)) โˆ’ ((๐ด ยทih ๐ท) + (๐ต ยทih ๐ถ)))
 
Theoremnormlem7tALT 30639 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   ((๐‘† โˆˆ โ„‚ โˆง (absโ€˜๐‘†) = 1) โ†’ (((โˆ—โ€˜๐‘†) ยท (๐ด ยทih ๐ต)) + (๐‘† ยท (๐ต ยทih ๐ด))) โ‰ค (2 ยท ((โˆšโ€˜(๐ต ยทih ๐ต)) ยท (โˆšโ€˜(๐ด ยทih ๐ด)))))
 
Theorembcseqi 30640 Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 30700. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   (((๐ด ยทih ๐ต) ยท (๐ต ยทih ๐ด)) = ((๐ด ยทih ๐ด) ยท (๐ต ยทih ๐ต)) โ†” ((๐ต ยทih ๐ต) ยทโ„Ž ๐ด) = ((๐ด ยทih ๐ต) ยทโ„Ž ๐ต))
 
Theoremnormlem9at 30641 Lemma used to derive properties of norm. Part of Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ ((๐ด โˆ’โ„Ž ๐ต) ยทih (๐ด โˆ’โ„Ž ๐ต)) = (((๐ด ยทih ๐ด) + (๐ต ยทih ๐ต)) โˆ’ ((๐ด ยทih ๐ต) + (๐ต ยทih ๐ด))))
 
20.2.2  Norms
 
Theoremdfhnorm2 30642 Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
normโ„Ž = (๐‘ฅ โˆˆ โ„‹ โ†ฆ (โˆšโ€˜(๐‘ฅ ยทih ๐‘ฅ)))
 
Theoremnormf 30643 The norm function maps from Hilbert space to reals. (Contributed by NM, 6-Sep-2007.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
normโ„Ž: โ„‹โŸถโ„
 
Theoremnormval 30644 The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of ๐ด is usually written as "|| ๐ด ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ (normโ„Žโ€˜๐ด) = (โˆšโ€˜(๐ด ยทih ๐ด)))
 
Theoremnormcl 30645 Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ (normโ„Žโ€˜๐ด) โˆˆ โ„)
 
Theoremnormge0 30646 The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ 0 โ‰ค (normโ„Žโ€˜๐ด))
 
Theoremnormgt0 30647 The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ (๐ด โ‰  0โ„Ž โ†” 0 < (normโ„Žโ€˜๐ด)))
 
Theoremnorm0 30648 The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(normโ„Žโ€˜0โ„Ž) = 0
 
Theoremnorm-i 30649 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ ((normโ„Žโ€˜๐ด) = 0 โ†” ๐ด = 0โ„Ž))
 
Theoremnormne0 30650 A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ ((normโ„Žโ€˜๐ด) โ‰  0 โ†” ๐ด โ‰  0โ„Ž))
 
Theoremnormcli 30651 Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    โ‡’   (normโ„Žโ€˜๐ด) โˆˆ โ„
 
Theoremnormsqi 30652 The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    โ‡’   ((normโ„Žโ€˜๐ด)โ†‘2) = (๐ด ยทih ๐ด)
 
Theoremnorm-i-i 30653 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    โ‡’   ((normโ„Žโ€˜๐ด) = 0 โ†” ๐ด = 0โ„Ž)
 
Theoremnormsq 30654 The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ ((normโ„Žโ€˜๐ด)โ†‘2) = (๐ด ยทih ๐ด))
 
Theoremnormsub0i 30655 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   ((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)) = 0 โ†” ๐ด = ๐ต)
 
Theoremnormsub0 30656 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ ((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)) = 0 โ†” ๐ด = ๐ต))
 
Theoremnorm-ii-i 30657 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   (normโ„Žโ€˜(๐ด +โ„Ž ๐ต)) โ‰ค ((normโ„Žโ€˜๐ด) + (normโ„Žโ€˜๐ต))
 
Theoremnorm-ii 30658 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (normโ„Žโ€˜(๐ด +โ„Ž ๐ต)) โ‰ค ((normโ„Žโ€˜๐ด) + (normโ„Žโ€˜๐ต)))
 
Theoremnorm-iii-i 30659 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‹    โ‡’   (normโ„Žโ€˜(๐ด ยทโ„Ž ๐ต)) = ((absโ€˜๐ด) ยท (normโ„Žโ€˜๐ต))
 
Theoremnorm-iii 30660 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹) โ†’ (normโ„Žโ€˜(๐ด ยทโ„Ž ๐ต)) = ((absโ€˜๐ด) ยท (normโ„Žโ€˜๐ต)))
 
Theoremnormsubi 30661 Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)) = (normโ„Žโ€˜(๐ต โˆ’โ„Ž ๐ด))
 
Theoremnormpythi 30662 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   ((๐ด ยทih ๐ต) = 0 โ†’ ((normโ„Žโ€˜(๐ด +โ„Ž ๐ต))โ†‘2) = (((normโ„Žโ€˜๐ด)โ†‘2) + ((normโ„Žโ€˜๐ต)โ†‘2)))
 
Theoremnormsub 30663 Swapping order of subtraction doesn't change the norm of a vector. (Contributed by NM, 14-Aug-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)) = (normโ„Žโ€˜(๐ต โˆ’โ„Ž ๐ด)))
 
Theoremnormneg 30664 The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
(๐ด โˆˆ โ„‹ โ†’ (normโ„Žโ€˜(-1 ยทโ„Ž ๐ด)) = (normโ„Žโ€˜๐ด))
 
Theoremnormpyth 30665 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ ((๐ด ยทih ๐ต) = 0 โ†’ ((normโ„Žโ€˜(๐ด +โ„Ž ๐ต))โ†‘2) = (((normโ„Žโ€˜๐ด)โ†‘2) + ((normโ„Žโ€˜๐ต)โ†‘2))))
 
Theoremnormpyc 30666 Corollary to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ ((๐ด ยทih ๐ต) = 0 โ†’ (normโ„Žโ€˜๐ด) โ‰ค (normโ„Žโ€˜(๐ด +โ„Ž ๐ต))))
 
Theoremnorm3difi 30667 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‹    โ‡’   (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)) โ‰ค ((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ถ)) + (normโ„Žโ€˜(๐ถ โˆ’โ„Ž ๐ต)))
 
Theoremnorm3adifii 30668 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‹    โ‡’   (absโ€˜((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ถ)) โˆ’ (normโ„Žโ€˜(๐ต โˆ’โ„Ž ๐ถ)))) โ‰ค (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต))
 
Theoremnorm3lem 30669 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‹    &   ๐ท โˆˆ โ„    โ‡’   (((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ถ)) < (๐ท / 2) โˆง (normโ„Žโ€˜(๐ถ โˆ’โ„Ž ๐ต)) < (๐ท / 2)) โ†’ (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)) < ๐ท)
 
Theoremnorm3dif 30670 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 20-Apr-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)) โ‰ค ((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ถ)) + (normโ„Žโ€˜(๐ถ โˆ’โ„Ž ๐ต))))
 
Theoremnorm3dif2 30671 Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)) โ‰ค ((normโ„Žโ€˜(๐ถ โˆ’โ„Ž ๐ด)) + (normโ„Žโ€˜(๐ถ โˆ’โ„Ž ๐ต))))
 
Theoremnorm3lemt 30672 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
(((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โˆง (๐ถ โˆˆ โ„‹ โˆง ๐ท โˆˆ โ„)) โ†’ (((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ถ)) < (๐ท / 2) โˆง (normโ„Žโ€˜(๐ถ โˆ’โ„Ž ๐ต)) < (๐ท / 2)) โ†’ (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)) < ๐ท))
 
Theoremnorm3adifi 30673 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 3-Oct-1999.) (New usage is discouraged.)
๐ถ โˆˆ โ„‹    โ‡’   ((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (absโ€˜((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ถ)) โˆ’ (normโ„Žโ€˜(๐ต โˆ’โ„Ž ๐ถ)))) โ‰ค (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)))
 
Theoremnormpari 30674 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   (((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต))โ†‘2) + ((normโ„Žโ€˜(๐ด +โ„Ž ๐ต))โ†‘2)) = ((2 ยท ((normโ„Žโ€˜๐ด)โ†‘2)) + (2 ยท ((normโ„Žโ€˜๐ต)โ†‘2)))
 
Theoremnormpar 30675 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต))โ†‘2) + ((normโ„Žโ€˜(๐ด +โ„Ž ๐ต))โ†‘2)) = ((2 ยท ((normโ„Žโ€˜๐ด)โ†‘2)) + (2 ยท ((normโ„Žโ€˜๐ต)โ†‘2))))
 
Theoremnormpar2i 30676 Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‹    โ‡’   ((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต))โ†‘2) = (((2 ยท ((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ถ))โ†‘2)) + (2 ยท ((normโ„Žโ€˜(๐ต โˆ’โ„Ž ๐ถ))โ†‘2))) โˆ’ ((normโ„Žโ€˜((๐ด +โ„Ž ๐ต) โˆ’โ„Ž (2 ยทโ„Ž ๐ถ)))โ†‘2))
 
Theorempolid2i 30677 Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‹    &   ๐ท โˆˆ โ„‹    โ‡’   (๐ด ยทih ๐ต) = (((((๐ด +โ„Ž ๐ถ) ยทih (๐ท +โ„Ž ๐ต)) โˆ’ ((๐ด โˆ’โ„Ž ๐ถ) ยทih (๐ท โˆ’โ„Ž ๐ต))) + (i ยท (((๐ด +โ„Ž (i ยทโ„Ž ๐ถ)) ยทih (๐ท +โ„Ž (i ยทโ„Ž ๐ต))) โˆ’ ((๐ด โˆ’โ„Ž (i ยทโ„Ž ๐ถ)) ยทih (๐ท โˆ’โ„Ž (i ยทโ„Ž ๐ต)))))) / 4)
 
Theorempolidi 30678 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of Axiom ax-his3 30604. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   (๐ด ยทih ๐ต) = (((((normโ„Žโ€˜(๐ด +โ„Ž ๐ต))โ†‘2) โˆ’ ((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต))โ†‘2)) + (i ยท (((normโ„Žโ€˜(๐ด +โ„Ž (i ยทโ„Ž ๐ต)))โ†‘2) โˆ’ ((normโ„Žโ€˜(๐ด โˆ’โ„Ž (i ยทโ„Ž ๐ต)))โ†‘2)))) / 4)
 
Theorempolid 30679 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of Axiom ax-his3 30604. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (๐ด ยทih ๐ต) = (((((normโ„Žโ€˜(๐ด +โ„Ž ๐ต))โ†‘2) โˆ’ ((normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต))โ†‘2)) + (i ยท (((normโ„Žโ€˜(๐ด +โ„Ž (i ยทโ„Ž ๐ต)))โ†‘2) โˆ’ ((normโ„Žโ€˜(๐ด โˆ’โ„Ž (i ยทโ„Ž ๐ต)))โ†‘2)))) / 4))
 
20.2.3  Relate Hilbert space to normed complex vector spaces
 
Theoremhilablo 30680 Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
+โ„Ž โˆˆ AbelOp
 
Theoremhilid 30681 The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
(GIdโ€˜ +โ„Ž ) = 0โ„Ž
 
Theoremhilvc 30682 Hilbert space is a complex vector space. Vector addition is +โ„Ž, and scalar product is ยทโ„Ž. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
โŸจ +โ„Ž , ยทโ„Ž โŸฉ โˆˆ CVecOLD
 
Theoremhilnormi 30683 Hilbert space norm in terms of vector space norm. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
โ„‹ = (BaseSetโ€˜๐‘ˆ)    &    ยทih = (ยท๐‘–OLDโ€˜๐‘ˆ)    &   ๐‘ˆ โˆˆ NrmCVec    โ‡’   normโ„Ž = (normCVโ€˜๐‘ˆ)
 
Theoremhilhhi 30684 Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
โ„‹ = (BaseSetโ€˜๐‘ˆ)    &    +โ„Ž = ( +๐‘ฃ โ€˜๐‘ˆ)    &    ยทโ„Ž = ( ยท๐‘ OLD โ€˜๐‘ˆ)    &    ยทih = (ยท๐‘–OLDโ€˜๐‘ˆ)    &   ๐‘ˆ โˆˆ NrmCVec    โ‡’   ๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ
 
Theoremhhnv 30685 Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    โ‡’   ๐‘ˆ โˆˆ NrmCVec
 
Theoremhhva 30686 The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    โ‡’    +โ„Ž = ( +๐‘ฃ โ€˜๐‘ˆ)
 
Theoremhhba 30687 The base set of Hilbert space. This theorem provides an independent proof of df-hba 30489 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    โ‡’    โ„‹ = (BaseSetโ€˜๐‘ˆ)
 
Theoremhh0v 30688 The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    โ‡’   0โ„Ž = (0vecโ€˜๐‘ˆ)
 
Theoremhhsm 30689 The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    โ‡’    ยทโ„Ž = ( ยท๐‘ OLD โ€˜๐‘ˆ)
 
Theoremhhvs 30690 The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    โ‡’    โˆ’โ„Ž = ( โˆ’๐‘ฃ โ€˜๐‘ˆ)
 
Theoremhhnm 30691 The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    โ‡’   normโ„Ž = (normCVโ€˜๐‘ˆ)
 
Theoremhhims 30692 The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    &   ๐ท = (normโ„Ž โˆ˜ โˆ’โ„Ž )    โ‡’   ๐ท = (IndMetโ€˜๐‘ˆ)
 
Theoremhhims2 30693 Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    &   ๐ท = (IndMetโ€˜๐‘ˆ)    โ‡’   ๐ท = (normโ„Ž โˆ˜ โˆ’โ„Ž )
 
Theoremhhmet 30694 The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    &   ๐ท = (IndMetโ€˜๐‘ˆ)    โ‡’   ๐ท โˆˆ (Metโ€˜ โ„‹)
 
Theoremhhxmet 30695 The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    &   ๐ท = (IndMetโ€˜๐‘ˆ)    โ‡’   ๐ท โˆˆ (โˆžMetโ€˜ โ„‹)
 
Theoremhhmetdval 30696 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    &   ๐ท = (IndMetโ€˜๐‘ˆ)    โ‡’   ((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (๐ด๐ท๐ต) = (normโ„Žโ€˜(๐ด โˆ’โ„Ž ๐ต)))
 
Theoremhhip 30697 The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    โ‡’    ยทih = (ยท๐‘–OLDโ€˜๐‘ˆ)
 
Theoremhhph 30698 The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
๐‘ˆ = โŸจโŸจ +โ„Ž , ยทโ„Ž โŸฉ, normโ„ŽโŸฉ    โ‡’   ๐‘ˆ โˆˆ CPreHilOLD
 
20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality
 
TheorembcsiALT 30699 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   (absโ€˜(๐ด ยทih ๐ต)) โ‰ค ((normโ„Žโ€˜๐ด) ยท (normโ„Žโ€˜๐ต))
 
TheorembcsiHIL 30700 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Proved from ZFC version.) (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   (absโ€˜(๐ด ยทih ๐ต)) โ‰ค ((normโ„Žโ€˜๐ด) ยท (normโ„Žโ€˜๐ต))
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