Metamath Proof Explorer

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Metamath Proof Explorer (1-4957)

Hilbert Space Explorer (4958-5787)

**Statement List for Metamath Proof Explorer -** 5301-5400 - Page 54 of 58
Type	Label	Description
Statement

Theorem	hsupval2t 5301	Value of supremum of set of subsets of Hilbert space. Definition of supremum in Proposition 1 of [Kalmbach] p. 65.
$\/H$

Theorem	hsupvalt 5302	Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2t 5301.
$\/H$

Theorem	chsupval2t 5303	The value of the supremum of a set of closed subspaces of Hilbert space. Definition of supremum in Proposition 1 of [Kalmbach] p. 65.
$\/H$

Theorem	chsupvalt 5304	The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2t 5303.
$\/H$

Theorem	spanclt 5305	The span of a subset of Hilbert space is a subspace.


Theorem	elspanclt 5306	A member of a span is a vector.
$/\$

Theorem	shsupclt 5307	Closure of the subspace supremum of set of subsets of Hilbert space.


Theorem	hsupclt 5308	Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to even if the subsets do not.
$\/H$

Theorem	chsupclt 5309	Closure of supremum of subset of . Definition of supremum in Proposition 1 of [Kalmbach] p. 65. Shows that is a complete lattice.
$\/H$

Theorem	hsupss 5310	Subset relation for supremum of Hilbert space subsets.
$/\$ $\/H$ $\/H$

Theorem	chsupss 5311	Subset relation for supremum of subset of .
$/\$ $\/H$ $\/H$

Theorem	chsupid 5312	A subspace is the supremum of all smaller subspaces.
$\/H$

Theorem	chsupsn 5313	Value of supremum of subset of on a singleton.
$\/H$

Theorem	hsupunss 5314	The union of a set of Hilbert space subsets is smaller than its supremum.
$\/H$

Theorem	spanss2 5315	A subset of Hilbert space is included in its span.


Theorem	shsupunss 5316	The union of a set of subspaces is smaller than its supremum.


Theorem	chsupunss 5317	The union of a set of closed subspaces is smaller than its supremum.
$\/H$

Theorem	spanid 5318	A subspace of Hilbert space is its own span.


Theorem	spanss 5319	Ordering relationship for the spans of subsets of Hilbert space.
$/\$

Theorem	spanssoc 5320	The span of a subset of Hilbert space is less than or equal to its closure (double orthogonal complement).


Theorem	sshjvalt 5321	Value of join for subsets of Hilbert space.
$/\$

Theorem	shjvalt 5322	Value of join in .
$/\$

Theorem	chjvalt 5323	Value of join in .
$/\$

Theorem	chjval 5324	Value of join in .


Theorem	dfchj2 5325	Alternate definition of join in the set of closed subspaces of Hilbert space .
$/\$ $/\$

Theorem	dfchj3 5326	Alternate definition of join in the set of closed subspaces of Hilbert space : the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice .
$/\$ $/\$ $\/H$

Theorem	sshjval3t 5327	Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3.
$/\$ $\/H$

Theorem	sshjclt 5328	Closure of join for subsets of Hilbert space.
$/\$

Theorem	shjclt 5329	Closure of join in .
$/\$

Theorem	chjclt 5330	Closure of join in .
$/\$

Theorem	shjcomt 5331	Commutative law for Hilbert lattice join of subspaces.
$/\$

Theorem	shincl 5332	Closure of intersection of two subspaces.


Theorem	shscom 5333	Commutative law for subspace sum.


Theorem	shsva 5334	Vector sum belongs to subspace sum.
$/\$

Theorem	shsel1 5335	A subspace sum contains a member of one of its subspaces.


Theorem	shsel2 5336	A subspace sum contains a member of one of its subspaces.


Theorem	shsvs 5337	Vector subtraction belongs to subspace sum.
$/\$

Theorem	shunss 5338	Union is smaller than subspace sum.


Theorem	shslej 5339	Subspace sum is smaller than Hilbert lattice join. Remark of [Kalmbach] p. 65.


Theorem	shunssj 5340	Union is smaller than Hilbert lattice join.


Theorem	shjcom 5341	Commutative law for join in .


Theorem	shsub1 5342	Subspace sum is an upper bound of its arguments.


Theorem	shsub2 5343	Subspace sum is an upper bound of its arguments.


Theorem	shub1 5344	Hilbert lattice join is an upper bound of two subspaces.


Theorem	shjcl 5345	Closure of join.


Theorem	shjshcl 5346	closure of join.


Theorem	shlub 5347	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces.
$/\$

Theorem	shless 5348	Subset implies subset of subspace sum.


Theorem	shlej1 5349	Add disjunct to both sides of Hilbert subspace ordering.


Theorem	shlej2 5350	Add disjunct to both sides of Hilbert subspace ordering.


Theorem	shslejt 5351	Subspace sum is smaller than subspace join. Remark of [Kalmbach] p. 65.
$/\$

Theorem	shinclt 5352	Closure of intersection of two subspaces.
$/\$

Theorem	shub1t 5353	Hilbert lattice join is an upper bound of two subspaces.
$/\$

Theorem	shub2t 5354	A subspace is a subset of its Hilbert lattice join with another.
$/\$

Theorem	shlubt 5355	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces.
$/\$ $/\$ $/\$

Theorem	shlej1t 5356	Add disjunct to both sides of Hilbert subspace ordering.
$/\$ $/\$

Theorem	shlej2t 5357	Add disjunct to both sides of Hilbert subspace ordering.
$/\$ $/\$

Theorem	shsidm 5358	Idempotent law for Hilbert subspace sum.


Theorem	shslub 5359	Least upper bound law for Hilbert subspace sum.
$/\$

Theorem	shlesb1 5360	Hilbert lattice ordering in terms of subspace sum.


Theorem	shsumval2 5361	An alternate way to express subspace sum.


Theorem	shsumval3 5362	An alternate way to express subspace sum.


Theorem	shmods 5363	The modular law holds for subspace sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70.


Theorem	shmod 5364	The modular law is implied by the closure of subspace sum. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70.