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**Statement List for Metamath Proof Explorer -** 9401-9500 - Page 95 of 123
Type	Label	Description
Statement

Definition	df-ch0 9401	Define the zero for closed subspaces of Hilbert space. See h0elch 9403 for closure law.


Theorem	elch0 9402	Membership in zero for closed subspaces of Hilbert space.


Theorem	h0elch 9403	The zero subspace is a closed subspace. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	h0elsh 9404	The zero subspace is a subspace of Hilbert space.


Theorem	hhssva 9405	The vector addition operation on a subspace.


Theorem	hhsssm 9406	The scalar multiplication operation on a subspace.


Theorem	hhssnm 9407	The norm operation on a subspace.


Theorem	hhssabli 9408	Abelian group property of subspace addition.
Abel

Theorem	hhssabl 9409	Abelian group property of subspace addition.
Abel

Theorem	hhssnv 9410	Normed complex vector space property of a subspace.
NrmCVec

Theorem	hhssnvt 9411	Normed complex vector space property of a subspace.
NrmCVec

Theorem	hhsst 9412	A member of is a subspace.
SubSp

Theorem	hhshsslem1 9413	Lemma for hhsssh 9415.

Theorem	hhshsslem2 9414	Lemma for hhsssh 9415.

Theorem	hhsssh 9415	The predicate " is a subspace of Hilbert space."
SubSp $/\$

Theorem	hhsssh2 9416	The predicate " is a subspace of Hilbert space."
NrmCVec $/\$

Theorem	hhssba 9417	The base set of a subspace.
BaseSet

Theorem	hhssvs 9418	The vector subtraction operation on a subspace.


Theorem	hhssvsf 9419	Mapping of the vector subtraction operation on a subspace.


Theorem	hhssph 9420	Inner product space property of a subspace.
CPreHil

Theorem	hhssims 9421	Induced metric of a subspace.
IndMet

Theorem	hhssims2 9422	Induced metric of a subspace.
IndMet

Theorem	hhssmet 9423	Induced metric of a subspace.
IndMet Met

Theorem	hhssmetba 9424	The base set for the metric of a subspace.
IndMet

Theorem	hhssmetdval 9425	Value of the distance function of the metric space of a subspace.
IndMet $/\$

Theorem	hhsscms 9426	The induced metric of a closed subspace is complete.
IndMet CMet

Theorem	hhssbn 9427	Banach space property of a closed subspace.
CBan

Theorem	hhsshl 9428	Hilbert space property of a closed subspace.
CHil

Theorem	ocval 9429	Value of orthogonal complement of a subset of Hilbert space.


Theorem	ocel 9430	Membership in orthogonal complement of H subset.
$/\$

Theorem	shocel 9431	Membership in orthogonal complement of H subspace.
$/\$

Theorem	ocsh 9432	The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107.


Theorem	shocsh 9433	The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107.


Theorem	ocss 9434	An orthogonal complement is a subset of Hilbert space.


Theorem	shocss 9435	An orthogonal complement is a subset of Hilbert space.


Theorem	occon 9436	Contraposition law for orthogonal complement.
$/\$

Theorem	occon2 9437	Double contraposition for orthogonal complement.
$/\$

Theorem	occon2i 9438	Double contraposition for orthogonal complement.


Theorem	oc0 9439	The zero vector belongs to an orthogonal complement of a Hilbert subspace.


Theorem	ocorth 9440	Members of a subset and its complement are orthogonal.
$/\$

Theorem	shocorth 9441	Members of a subspace and its complement are orthogonal.
$/\$

Theorem	ococss 9442	Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	shococss 9443	Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	shorth 9444	Members of orthogonal subspaces are orthogonal.
$/\$

Theorem	ocin 9445	Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	ocnel 9446	A nonzero vector in the complement of a subspace does not belong to the subspace.
$/\$ $/\$

Theorem	chocvali 9447	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	chocunii 9448	Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part).

Theorem	occllem1 9449	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem2 9450	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem3 9451	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem4 9452	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem5 9453	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem6 9454	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem7 9455	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem8 9456	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occli 9457	Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107.


Theorem	occl 9458	Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107.


Theorem	shoccl 9459	Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.


Theorem	choccl 9460	Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.


Theorem	choccli 9461	Closure of orthocomplement.


Projection theorem

Theorem	projlem1 9462	Part of Lemma 3.6 of [Beran] p. 100: "Choose e > 0. Let n₀ be a natural number satisfying the inequality n₀ > 4(2i₀ + 1) e -1." Used by projlem2 9463.


Theorem	projlem2 9463	Part of Lemma 3.6 of [Beran] p. 100. We need the square root for the norm limit. Used by projlem28 9489.


Theorem	projlem3 9464	Part of Lemma 3.6 of [Beran] p. 100, bottom inequality. Used by projlem6 9467.


Theorem	projlem4 9465	Part of Lemma 3.6 of [Beran] p. 101, top. Used by projlem6 9467.
$/\$

Theorem	projlem5 9466	Part of Lemma 3.6 of [Beran] p. 100, bottom. Used by projlem6 9467.


Theorem	projlem6 9467	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem7 9468.
$/\$

Theorem	projlem7 9468	Part of Lemma 3.6 of [Beran] p. 101. Applies weak deduction theorem to projlem6 9467. Used by projlem19 9480.
$/\$ $/\$

Theorem	projlem8 9469	Part of Lemma 3.6 of [Beran] p. 100. The set is a non-empty set of reals with an upper bound. Part of Lemma 3.6 of [Beran] p. 100. Used by projlem9 9470 projlem12 9473 projlem13 9474 projlem15 9476. Note we use 'supremum'; its negative is the infimum.
$/\$ $/\$

Theorem	projlem9 9470	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Real closure of the infimum of norms. Used by projlem11 9472 projlem12 9473 projlem13 9474 projlem15 9476.


Theorem	projlem10 9471	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). A member of the infimum set. Used by projlem12 9473.


Theorem	projlem11 9472	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). is the infimum of the set of norms. Show it is real. Used by projlem12 9473 projlem13 9474 projlem14 9475 projlem15 9476 projlem18 9479 projlem19 9480 projlem26 9487 projlem28 9489 projlem31 9492.


Theorem	projlem12 9473	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). The infimum is less than any norm in the set of norms. Used by projlem14 9475 projlem18 9479 projlem31 9492.


Theorem	projlem13 9474	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). The infimum of the set of norms is nonnegative. Used by projlem18 9479 projlem19 9480 projlem28 9489.


Theorem	projlem14 9475	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Used by projlem16 9477.


Theorem	projlem15 9476	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Used by projlem16 9477.


Theorem	projlem16 9477	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Existence of a vector sequence member, used to show (via Axiom of Choice) the vector sequence existence. Used by projlem17 9478.
$/\$

Theorem	projlem17 9478	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). This uses the Axiom of Choice to show the existence of a vector sequence satisfying the assumption of Lemma 3.6 of [Beran] p. 100: "Let {y_n } be a sequence of W such that i₀ - 1/n < \|\|x₀ - y_n \|\| < i₀ + 1/n." Here, corresponds to "W"; to "{y_n }"; to "n"; to "i₀ "; and to "\|\|x₀ - y_n \|\|". Used by projlem 9493.
$/\$ $/\$

Theorem	projlem18 9479	Part of Lemma 3.6 of [Beran] p. 101, top. Used by projlem19 9480.


Theorem	projlem19 9480	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem20 9481.
$/\$ $/\$

Theorem	projlem20 9481	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem27 9488.
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	projlem21 9482	Part of Lemma 3.6 of [Beran] p. 100. The hypothesis lets us work with our postulated vector sequence (whose existence was shown by projlem17 9478). Here we just show the sequence value belongs to the closed subspace . Used by projlem27 9488 projlem28 9489.
$/\$ $/\$

Theorem	projlem22 9483	Part of Lemma 3.6 of [Beran] p. 100. Here we show a member of the vector sequence is bounded. Used by projlem27 9488.
$/\$ $/\$

Theorem	projlem23 9484	Part of Lemma 3.6 of [Beran] p. 101. The hypothesis lets us work with the sequence which corresponds to Beran's "{\|\|yn-x0\|\|}". Used by projlem25 9486 projlem26 9487.
$/\$

Theorem	projlem24 9485	Part of Lemma 3.6 of [Beran] p. 101. Here we show our vector sequence implies the real numbers sequence corresponding to Beran's "{\|\|yn-x0\|\|}". Used by projlem25 9486 projlem26 9487.
$/\$

Theorem	projlem25 9486	Part of Lemma 3.6 of [Beran] p. 101. "The sequence {\|\|yn-x0\|\|} of real numbers converges to the number \|\|y0-x0\|\|" (here, "y0" is and "x0" is ). Used by projlem31 9492.
$/\$

Theorem	projlem26 9487	Part of Lemma 3.6 of [Beran] p. 101. The real sequence (Beran's "{\|\|yn-x0\|\|}") converges to the infimum of norms. Used by projlem31 9492.
$/\$ $/\$ $/\$

Theorem	projlem27 9488	Part of Lemma 3.6 of [Beran] p. 101. Boundedness to show is a Cauchy sequence. Used by projlem28 9489.
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	projlem28 9489	Part of Lemma 3.6 of [Beran] p. 101. Boundedness to show is a Cauchy sequence. Used by projlem29 9490.
$/\$ $/\$ $/\$

Theorem	projlem29 9490	Part of Lemma 3.6 of [Beran] p. 101: 'Hence, {yn} is a Cauchy sequence.' Used by projlem30 9491.
$/\$ $/\$

Theorem	projlem30 9491	Part of Lemma 3.6 of [Beran] p. 101: 'By completeness of W, there exists y0 e. W such that yn->y0.' Used by projlem31 9492.
$/\$ $/\$

Theorem	projlem31 9492	Part of Lemma 3.6 of [Beran] p. 101. The postulated vector sequence implies our conclusion. By showing such a sequence exists (which was done with the Axiom of Choice in projlem17 9478), we can show the final conclusion, projlem 9493.
$/\$ $/\$

Theorem	projlem 9493	Lemma 3.6 of [Beran] p. 101: "Let be a complete subspace of a (pre-)Hilbert space and let . Then there exists a vector such that for every ." This is a lemma for the projection theorem.


Theorem	projlemHIL 9494	Lemma 3.6 of [Beran] p. 101: "Let be a complete subspace of a (pre-)Hilbert space and let . Then there exists a vector such that for every ." This is a lemma for the projection theorem.


Theorem	pjthlem1 9495	Lemma for pjthi 9509.

Theorem	pjthlem2 9496	Lemma for pjthi 9509.

Theorem	pjthlem3 9497	Lemma for pjthi 9509.

Theorem	pjthlem4 9498	Lemma for pjthi 9509.

Theorem	pjthlem5 9499	Lemma for pjthi 9509.

Theorem	pjthlem6 9500	Lemma for pjthi 9509.

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