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Theorem List for Metamath Proof Explorer - 18701-18800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnmofval 18701* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoval 18702* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmogelb 18703* Property of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmolb 18704* Any upper bound on the values of a linear operator translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmolb2d 18705* Any upper bound on the values of a linear operator at nonzero vectors translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp       NrmGrp

Theoremnmof 18706 The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmocl 18707 The operator norm of an operator is an extended real. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoge0 18708 The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnghmfval 18709 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremisnghm 18710 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp NrmGrp

Theoremisnghm2 18711 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremisnghm3 18712 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theorembddnghm 18713 A bounded group homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremnghmcl 18714 A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmoi 18715 The operator norm achieves the minimum of the set of upper bounds, if the operator is bounded. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmoix 18716 The operator norm is a bound on the size of an operator, even when it is infinite (using extended real multiplication). (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoi2 18717 The operator norm is a bound on the growth of a vector under the action of the operator. (Contributed by Mario Carneiro, 19-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoleub 18718* The operator norm, defined as an infimum of upper bounds, can also be defined as a supremum of norms of away from zero. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp       NrmGrp

Theoremnghmrcl1 18719 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp

Theoremnghmrcl2 18720 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp

Theoremnghmghm 18721 A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmo0 18722 The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoeq0 18723 The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoco 18724 An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom

Theoremnghmco 18725 The composition of normed group homomorphisms is a normed group homomorphism. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom NGHom

Theoremnmotri 18726 Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom

Theoremnghmplusg 18727 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom NGHom

Theorem0nghm 18728 The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremnmoid 18729 The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp

Theoremidnghm 18730 The identity operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NGHom

Theoremnmods 18731 Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.)
NGHom

Theoremnghmcn 18732 A normed group homomorphism is a continuous function. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom

Theoremisnmhm 18733 A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NrmMod NrmMod LMHom NGHom

Theoremnmhmrcl1 18734 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NrmMod

Theoremnmhmrcl2 18735 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NrmMod

Theoremnmhmlmhm 18736 A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom LMHom

Theoremnmhmnghm 18737 A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NGHom

Theoremnmhmghm 18738 A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom

Theoremisnmhm2 18739 A normed module homomorphism is a left module homomorphism with bounded norm (a bounded linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmMod NrmMod LMHom NMHom

Theoremnmhmcl 18740 A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom

Theoremidnmhm 18741 The identity operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmMod NMHom

Theorem0nmhm 18742 The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
Scalar       Scalar       NrmMod NrmMod NMHom

Theoremnmhmco 18743 The composition of bounded linear operators is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NMHom NMHom NMHom

Theoremnmhmplusg 18744 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
NMHom NMHom NMHom

11.4.10  Topology on the reals

Theoremqtopbaslem 18745 The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremqtopbas 18746 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)

Theoremretopbas 18747 A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)

Theoremretop 18748 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)

Theoremuniretop 18749 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)

Theoremretopon 18750 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
TopOn

TheoremretpsOLD 18751 The standard topological space on the reals. (Contributed by NM, 10-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremretps 18752 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
TopSet

Theoremiooretop 18753 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)

Theoremicccld 18754 Closed intervals are closed sets of the standard topology on . (Contributed by FL, 14-Sep-2007.)

Theoremicopnfcld 18755 Right-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremiocmnfcld 18756 Left-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremqdensere 18757 is dense in the standard topology on . (Contributed by NM, 1-Mar-2007.)

Theoremcnmetdval 18758 Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremcnmet 18759 The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)

Theoremcnxmet 18760 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)

Theoremcnbl0 18761 Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremcnblcld 18762* Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremcnfldms 18763 The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld

Theoremcnfldxms 18764 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld

Theoremcnfldtps 18765 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld

Theoremcnfldnm 18766 The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld

Theoremcnngp 18767 The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld NrmGrp

Theoremcnnrg 18768 The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld NrmRing

Theoremcnfldtopn 18769 The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld

Theoremcnfldtopon 18770 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
fld       TopOn

Theoremcnfldtop 18771 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
fld

Theoremcnfldhaus 18772 The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
fld

Theoremremetdval 18773 Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)

Theoremremet 18774 The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)

Theoremrexmet 18775 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)

Theorembl2ioo 18776 A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremioo2bl 18777 An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Theoremioo2blex 18778 An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremblssioo 18779 The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremtgioo 18780 The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremqdensere2 18781 is dense in . (Contributed by NM, 24-Aug-2007.)

Theoremblcvx 18782 An open ball in the complex numbers is a convex set. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremrehaus 18783 The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)

Theoremtgqioo 18784 The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremre2ndc 18785 The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremresubmet 18786 The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
t

Theoremtgioo2 18787 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)
fld       t

Theoremrerest 18788 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.)
fld              t t

Theoremtgioo3 18789 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)
flds

Theoremxrtgioo 18790 The topology on the extended reals coincides with the standard topology on the reals, when restricted to . (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop t

Theoremxrrest 18791 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop               t t

Theoremxrrest2 18792 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
fld       ordTop        t t

Theoremxrsxmet 18793 The metric on the extended reals is a proper extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremxrsdsre 18794 The metric on the extended reals coincides with the usual metric on the reals. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremxrsblre 18795 Any ball of the metric of the extended reals centered on an element of is entirely contained in . (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremxrsmopn 18796 The metric on the extended reals generates a topology, but this does not match the order topology on ; for example is open in the metric topology, but not the order topology. However, the metric topology is finer than the order topology, meaning that all open intervals are open in the metric topology. (Contributed by Mario Carneiro, 4-Sep-2015.)
ordTop

Theoremzcld 18797 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremrecld2 18798 The real numbers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
fld

Theoremzcld2 18799 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
fld

Theoremzdis 18800 The integers are a discrete set in the topology on . (Contributed by Mario Carneiro, 19-Sep-2015.)
fld       t

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