P. 281 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 28001-28100 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	normneg 28001	The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)


Theorem	normpyth 28002	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.)
$/\$

Theorem	normpyc 28003	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.)
$/\$

Theorem	norm3difi 28004	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.)


Theorem	norm3adifii 28005	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.)


Theorem	norm3lem 28006	Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
$/\$

Theorem	norm3dif 28007	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.)
$/\$ $/\$

Theorem	norm3dif2 28008	Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
$/\$ $/\$

Theorem	norm3lemt 28009	Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	norm3adifi 28010	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.)
$/\$

Theorem	normpari 28011	Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)


Theorem	normpar 28012	Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
$/\$

Theorem	normpar2i 28013	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.)


Theorem	polid2i 28014	Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)


Theorem	polidi 28015	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 27941. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)


Theorem	polid 28016	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 27941. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
$/\$

19.2.3 Relate Hilbert space to normed complex vector spaces

Theorem	hilablo 28017	Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)


Theorem	hilid 28018	The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
GId

Theorem	hilvc 28019	Hilbert space is a complex vector space. Vector addition is , and scalar product is . (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)


Theorem	hilnormi 28020	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

Theorem	hilhhi 28021	Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)


Theorem	hhnv 28022	Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)


Theorem	hhva 28023	The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)


Theorem	hhba 28024	The base set of Hilbert space. This theorem provides an independent proof of df-hba 27826 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)


Theorem	hh0v 28025	The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)


Theorem	hhsm 28026	The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)


Theorem	hhvs 28027	The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)


Theorem	hhnm 28028	The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
_CV

Theorem	hhims 28029	The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)


Theorem	hhims2 28030	Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)


Theorem	hhmet 28031	The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)


Theorem	hhxmet 28032	The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)


Theorem	hhmetdval 28033	Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
$/\$

Theorem	hhip 28034	The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)


Theorem	hhph 28035	The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)


19.2.4 Bunjakovaskij-Cauchy-Schwarz inequality

Theorem	bcsiALT 28036	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.)


Theorem	bcsiHIL 28037	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.)


Theorem	bcs 28038	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
$/\$

Theorem	bcs2 28039	Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28037. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
$/\$ $/\$

Theorem	bcs3 28040	Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28037. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
$/\$ $/\$

19.3 Cauchy sequences and completeness axiom

19.3.1 Cauchy sequences and limits

Theorem	hcau 28041*	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.)
$/\$

Theorem	hcauseq 28042	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.)


Theorem	hcaucvg 28043*	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.)
$/\$

Theorem	seq1hcau 28044*	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.)


Theorem	hlimi 28045*	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.)
$/\$ $/\$

Theorem	hlimseqi 28046	A sequence with a limit on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)


Theorem	hlimveci 28047	Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)


Theorem	hlimconvi 28048*	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.)
$/\$

Theorem	hlim2 28049*	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.)
$/\$

Theorem	hlimadd 28050*	Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)


19.3.2 Derivation of the completeness axiom from ZF set theory

Theorem	hilmet 28051	The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)


Theorem	hilxmet 28052	The Hilbert space norm determines a metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)


Theorem	hilmetdval 28053	Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
$/\$

Theorem	hilims 28054	Hilbert space distance metric. (Contributed by NM, 13-Sep-2007.) (New usage is discouraged.)


Theorem	hhcau 28055	The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)


Theorem	hhlm 28056	The limit sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)


Theorem	hhcmpl 28057*	Lemma used for derivation of the completeness axiom ax-hcompl 28059 from ZFC Hilbert space theory. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)


Theorem	hilcompl 28058*	Lemma used for derivation of the completeness axiom ax-hcompl 28059 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 27856; the 6th would be satisfied by eqid 2622; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 27771. (Contributed by NM, 13-Sep-2007.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)


19.3.3 Completeness postulate for a Hilbert space

Axiom	ax-hcompl 28059*	Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)


19.3.4 Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces

Theorem	hhcms 28060	The Hilbert space induced metric determines a complete metric space. (Contributed by NM, 10-Apr-2008.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)


Theorem	hhhl 28061	The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)


Theorem	hilcms 28062	The Hilbert space norm determines a complete metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)


Theorem	hilhl 28063	The Hilbert space of the Hilbert Space Explorer is a complex Hilbert space. (Contributed by Steve Rodriguez, 29-Apr-2007.) (New usage is discouraged.)


19.4 Subspaces and projections

19.4.1 Subspaces

Definition	df-sh 28064	Define the set of subspaces of a Hilbert space. See issh 28065 for its membership relation. Basically, a subspace is a subset of a Hilbert space that acts like a vector space. From Definition of [Beran] p. 95. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
$/\$ $/\$

Theorem	issh 28065	Subspace of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	issh2 28066*	Subspace of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. Definition of [Beran] p. 95. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	shss 28067	A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)


Theorem	shel 28068	A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
$/\$

Theorem	shex 28069	The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)


Theorem	shssii 28070	A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)


Theorem	sheli 28071	A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)


Theorem	shelii 28072	A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)


Theorem	sh0 28073	The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)


Theorem	shaddcl 28074	Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
$/\$ $/\$

Theorem	shmulcl 28075	Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
$/\$ $/\$

Theorem	issh3 28076*	Subspace of a Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
$/\$ $/\$

Theorem	shsubcl 28077	Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
$/\$ $/\$

19.4.2 Closed subspaces

Definition	df-ch 28078	Define the set of closed subspaces of a Hilbert space. A closed subspace is one in which the limit of every convergent sequence in the subspace belongs to the subspace. For its membership relation, see isch 28079. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 28080 and isch3 28098. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)


Theorem	isch 28079	Closed subspace of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
$/\$

Theorem	isch2 28080*	Closed subspace of a Hilbert space. Definition of [Beran] p. 107. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
$/\$ $/\$

Theorem	chsh 28081	A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)


Theorem	chsssh 28082	Closed subspaces are subspaces in a Hilbert space. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)


Theorem	chex 28083	The set of closed subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)


Theorem	chshii 28084	A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)


Theorem	ch0 28085	The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)


Theorem	chss 28086	A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)


Theorem	chel 28087	A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
$/\$

Theorem	chssii 28088	A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)


Theorem	cheli 28089	A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)


Theorem	chelii 28090	A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)


Theorem	chlimi 28091	The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)
$/\$ $/\$

Theorem	hlim0 28092	The zero sequence in Hilbert space converges to the zero vector. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)


Theorem	hlimcaui 28093	If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)


Theorem	hlimf 28094	Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)


Theorem	hlimuni 28095	A Hilbert space sequence converges to at most one limit. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 2-May-2015.) (New usage is discouraged.)
$/\$

Theorem	hlimreui 28096*	The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)


Theorem	hlimeui 28097*	The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)


Theorem	isch3 28098*	A Hilbert subspace is closed iff it is complete. A complete subspace is one in which every Cauchy sequence of vectors in the subspace converges to a member of the subspace (Definition of complete subspace in [Beran] p. 96). Remark 3.12 of [Beran] p. 107. (Contributed by NM, 24-Dec-2001.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
$/\$

Theorem	chcompl 28099*	Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.)
$/\$ $/\$

Theorem	helch 28100	The unit Hilbert lattice element (which is all of Hilbert space) belongs to the Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 6-Sep-1999.) (New usage is discouraged.)

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