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**Statement List for Metamath Proof Explorer -** 9101-9200 - Page 92 of 107
Type	Label	Description
Statement

Theorem	hhsssm 9101	The scalar multiplication operation on a subspace.


Theorem	hhssnm 9102	The norm operation on a subspace.


Theorem	hhssabl 9103	Abelian group property of subspace addition.
Abel

Theorem	hhssablt 9104	Abelian group property of subspace addition.
Abel

Theorem	hhssnv 9105	Normed complex vector space property of a subspace.
NrmCVec

Theorem	hhssnvt 9106	Normed complex vector space property of a subspace.
NrmCVec

Theorem	hhsst 9107	A member of is a subspace.
SubSp

Theorem	hhshsslem1 9108	Lemma for hhsssh 9110.

Theorem	hhshsslem2 9109	Lemma for hhsssh 9110.

Theorem	hhsssh 9110	The predicate " is a subspace of Hilbert space."
SubSp $/\$

Theorem	hhsssh2 9111	The predicate " is a subspace of Hilbert space."
NrmCVec $/\$

Theorem	hhssba 9112	The base set of a subspace.
Base

Theorem	hhssvs 9113	The vector subtraction operation on a subspace.


Theorem	hhssvsf 9114	Mapping of the vector subtraction operation on a subspace.


Theorem	hhssph 9115	Inner product space property of a subspace.
CPreHil

Theorem	hhssims 9116	Induced metric of a subspace.
IndMet

Theorem	hhssims2 9117	Induced metric of a subspace.
IndMet

Theorem	hhssmet 9118	Induced metric of a subspace.
IndMet Met

Theorem	hhssmetba 9119	The base set for the metric of a subspace.
IndMet

Theorem	hhssmetdval 9120	Value of the distance function of the metric space of a subspace.
IndMet $/\$

Theorem	hhsscms 9121	The induced metric of a closed subspace is complete.
IndMet CMet

Theorem	hhssbn 9122	Banach space property of a closed subspace.
CBan

Theorem	hhsshl 9123	Hilbert space property of a closed subspace.
CHil

Theorem	ocvalt 9124	Value of orthogonal complement of a subset of Hilbert space.


Theorem	ocelt 9125	Membership in orthogonal complement of H subset.
$/\$

Theorem	shocelt 9126	Membership in orthogonal complement of H subspace.
$/\$

Theorem	ocsh 9127	The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107.


Theorem	shocsh 9128	The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107.


Theorem	ocss 9129	An orthogonal complement is a subset of Hilbert space.


Theorem	shocss 9130	An orthogonal complement is a subset of Hilbert space.


Theorem	occont 9131	Contraposition law for orthogonal complement.
$/\$

Theorem	occon2t 9132	Double contraposition for orthogonal complement.
$/\$

Theorem	occon2 9133	Double contraposition for orthogonal complement.


Theorem	oc0 9134	The zero vector belongs to an orthogonal complement of a Hilbert subspace.


Theorem	ocorth 9135	Members of a subset and its complement are orthogonal.
$/\$

Theorem	shocorth 9136	Members of a subspace and its complement are orthogonal.
$/\$

Theorem	ococss 9137	Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	shococss 9138	Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	shorth 9139	Members of orthogonal subspaces are orthogonal.
$/\$

Theorem	ocin 9140	Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	ocnelt 9141	A nonzero vector in the complement of a subspace does not belong to the subspace.
$/\$ $/\$

Theorem	chocval 9142	Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of is the set of vectors that are orthogonal to all vectors in .


Theorem	chocuni 9143	Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part).

Theorem	occllem1 9144	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem2 9145	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem3 9146	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem4 9147	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem5 9148	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem6 9149	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem7 9150	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem8 9151	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occl 9152	Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107.


Theorem	occlt 9153	Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107.


Theorem	shocclt 9154	Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.


Theorem	chocclt 9155	Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.
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