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Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhis6 22601 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.)

Theoremhis1i 22602 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.)

Theoremabshicom 22603 Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)

Theoremhial0 22604* A vector whose inner product is always zero is zero. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremhial02 22605* A vector whose inner product is always zero is zero. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)

Theoremhisubcomi 22606 Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)

Theoremhi2eq 22607 Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)

Theoremhial2eq 22608* Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)

Theoremhial2eq2 22609* Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)

Theoremorthcom 22610 Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)

Theoremnormlem0 22611 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.)

Theoremnormlem1 22612 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.)

Theoremnormlem2 22613 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.)

Theoremnormlem3 22614 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.)

Theoremnormlem4 22615 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.)

Theoremnormlem5 22616 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.)

Theoremnormlem6 22617 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.)

Theoremnormlem7 22618 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.)

Theoremnormlem8 22619 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)

Theoremnormlem9 22620 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)

Theoremnormlem7tALT 22621 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.)

Theorembcseqi 22622 Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 22682. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)

Theoremnormlem9at 22623 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.)

18.2.2  Norms

Theoremdfhnorm2 22624 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.)

Theoremnormf 22625 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.)

Theoremnormval 22626 The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of is usually written as "|| ||", 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.)

Theoremnormcl 22627 Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

Theoremnormge0 22628 The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

Theoremnormgt0 22629 The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)

Theoremnorm0 22630 The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Theoremnorm-i 22631 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

Theoremnormne0 22632 A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Theoremnormcli 22633 Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)

Theoremnormsqi 22634 The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)

Theoremnorm-i-i 22635 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.)

Theoremnormsq 22636 The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)

Theoremnormsub0i 22637 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Theoremnormsub0 22638 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Theoremnorm-ii-i 22639 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)

Theoremnorm-ii 22640 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremnorm-iii-i 22641 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

Theoremnorm-iii 22642 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)

Theoremnormsubi 22643 Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)

Theoremnormpythi 22644 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.)

Theoremnormsub 22645 Swapping order of subtraction doesn't change the norm of a vector. (Contributed by NM, 14-Aug-1999.) (New usage is discouraged.)

Theoremnormneg 22646 The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Theoremnormpyth 22647 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.)

Theoremnormpyc 22648 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.)

Theoremnorm3difi 22649 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.)

Theoremnorm3adifii 22650 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.)

Theoremnorm3lem 22651 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Theoremnorm3dif 22652 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.)

Theoremnorm3dif2 22653 Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)

Theoremnorm3lemt 22654 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Theoremnorm3adifi 22655 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.)

Theoremnormpari 22656 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)

Theoremnormpar 22657 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)

Theoremnormpar2i 22658 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.)

Theorempolid2i 22659 Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)

Theorempolidi 22660 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 22586. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)

Theorempolid 22661 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 22586. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

18.2.3  Relate Hilbert space to normed complex vector spaces

Theoremhilablo 22662 Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)

Theoremhilid 22663 The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
GId

Theoremhilvc 22664 Hilbert space is a complex vector space. Vector addition is , and scalar product is . (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)

Theoremhilnormi 22665 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.)
CV

Theoremhilhhi 22666 Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhnv 22667 Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremhhva 22668 The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremhhba 22669 The base set of Hilbert space. This theorem provides an independent proof of df-hba 22472 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremhh0v 22670 The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhhsm 22671 The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremhhvs 22672 The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)

Theoremhhnm 22673 The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
CV

Theoremhhims 22674 The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremhhims2 22675 Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhmet 22676 The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhxmet 22677 The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremhhmetdval 22678 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhip 22679 The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhhph 22680 The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)

TheorembcsiALT 22681 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

TheorembcsiHIL 22682 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.)

Theorembcs 22683 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)

Theorembcs2 22684 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 22682. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)

Theorembcs3 22685 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 22682. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)

18.3  Cauchy sequences and completeness axiom

18.3.1  Cauchy sequences and limits

Theoremhcau 22686* Member of the set of Cauchy sequences on a Hilbert space. Definition for Cauchy sequence in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremhcauseq 22687 A Cauchy sequences on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremhcaucvg 22688* A Cauchy sequence on a Hilbert space converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremseq1hcau 22689* A sequence on a Hilbert space is a Cauchy sequence if it converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremhlimi 22690* Express the predicate: The limit of vector sequence in a Hilbert space is , i.e. converges to . This means that for any real , no matter how small, there always exists an integer such that the norm of any later vector in the sequence minus the limit is less than . Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremhlimseqi 22691 A sequence with a limit on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Theoremhlimveci 22692 Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Theoremhlimconvi 22693* Convergence of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremhlim2 22694* The limit of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremhlimadd 22695* Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

18.3.2  Derivation of the completeness axiom from ZF set theory

Theoremhilmet 22696 The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)

Theoremhilxmet 22697 The Hilbert space norm determines a metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremhilmetdval 22698 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)

Theoremhilims 22699 Hilbert space distance metric. (Contributed by NM, 13-Sep-2007.) (New usage is discouraged.)

Theoremhhcau 22700 The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)

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