Hilbert Flow Spaces with Operators over Topological Graphs

2017·0.28 MB·

by  Linfan Mao

International J.Math. Combin. Vol.4(2017), 19-45
Hilbert Flow Spaces with Operators over Topological Graphs
Linfan MAO
1. ChineseAcademyofMathematicsandSystemScience,Beijing100190,P.R.China
2. AcademyofMathematicalCombinatorics&Applications(AMCA),Colorado,USA
E-mail: [email protected]
Abstract: AcomplexsystemS consistsmcomponents,maybeinconsistencewithm≥2,
such as those of biological systems or generally, interaction systems and usually, a system
with contradictions, which implies that there are no a mathematical subfield applicable.
Then,how can we holdon its global and local behaviors or reality? Allofusknowthatthere
→−
always exists universalconnections between thingsin theworld, i.e., a topological graph G
→− →−
underlyingcomponentsinS. Wecantherebyestablishmathematicsovergraphs G ,G ,···
1 2
→− →−
by viewing labeling graphs GL1,GL2,··· to be globally mathematical elements, not only
1 2
game objects or combinatorial structures, which can be applied to characterize dynamic
→− →−
behaviorsofthesystemS ontimet. Formally,acontinuityflow GLisatopologicalgraph G
associated with a mapping L:(v,u)→L(v,u), 2 end-operators A+vu :L(v,u)→LA+vu(v,u)
andA+uv :L(u,v)→LA+uv(u,v)onaBanachspaceB overafieldF withL(v,u)=−L(u,v)
and A+vu(−L(v,u))=−LA+vu(v,u) for ∀(v,u)∈E →−G holding with continuity equations
(cid:16) (cid:17)
LA+vu(v,u)=L(v), ∀v∈V →−G .
u∈XNG(v) (cid:16) (cid:17)
Themain purposeofthispaperis toextendBanach orHilbert spaces toBanach or Hilbert
→− →−
continuity flow spaces over topological graphs G ,G ,··· and establish differentials on
1 2
continuityflowsforcharacterizing theirgloballynchangerate.oAfewwell-known resultssuch
asthoseofTaylorformula,L’Hospital’sruleonlimitationaregeneralizedtocontinuityflows,
and algebraic or differential flow equations are discussed in this paper. All of these results
form theelementarydifferentialtheoryoncontinuityflows,whichcontributesmathematical
combinatorics and can be used to characterizing the behavior of complex systems, particu-
larly, thesynchronization.
Key Words: Complex system, Smarandache multispace, continuity flow, Banach space,
Hilbert space, differential, Taylor formula, L’Hospital’s rule, mathematical combinatorics.
AMS(2010): 34A26, 35A08, 46B25, 92B05, 05C10, 05C21, 34D43, 51D20.
§1. Introduction
A Banach or Hilbert space is respectively a linear space A over a field R or C equipped with a
complete norm or inner product , , i.e., for every Cauchy sequence x in A, there
n
k·k h · · i { }
1ReceivedMay5,2017,Accepted November6,2017.
20 LinfanMAO
exists an element x in A such that
lim xn x A =0 or lim xn x,xn x A =0
n k − k n h − − i
→∞ →∞
and a topological graph ϕ(G) is an embedding of a graph G with vertex set V(G), edge set
E(G) in a space S, i.e., there is a 1 1 continuous mapping ϕ : G ϕ(G) S with
− → ⊂
ϕ(p)=ϕ(q)ifp=q for p,q G,i.e.,edgesofGonlyintersectatverticesinS,anembedding
6 6 ∀ ∈
of a topological space to another space. A well-known result on embedding of graphs without
loops and multiple edges in Rn concluded that there always exists an embedding of G that all
edges are straight segments in Rn for n 3 ([22]) such as those shown in Fig.1.
≥
Fig.1
Asweknown,thepurposeofscienceisholdontherealityofthingsinthe world. However,
the reality of a thing T is complex and there are no a mathematical subfield applicable unless
a system maybe with contradictions in general. Is such a contradictory system meaningless
to human beings? Certain not because all of these contradictions are the result of human
beings, not the nature of things themselves, particularly on those of contradictory systems in
mathematics. Thus,holdingontherealityofthingsmotivatesonetoturncontradictorysystems
to compatibleone bya combinatorialnotionandestablishanenvelopetheoryonmathematics,
i.e.,mathematicalcombinatorics([9]-[13]). Then,Can weglobally characterize thebehavior of a
systemorapopulation withelements 2,whichmaybecontradictoryorcompatible? Theanswer
≥
is certainly YES by continuity flows, which needs one to establish an envelope mathematical
theory over topological graphs, i.e., views labeling graphs GL to be mathematical elements
([19]), not only a game object or a combinatorial structure with labels in the following sense.
Definition 1.1 A continuity flow −→G;L,A is an oriented embedded graph −→G in a topological
space S associated with a mappin(cid:16)g L : v (cid:17) L(v), (v,u) L(v,u), 2 end-operators A+ :
→ → vu
L(v,u)→LA+vu(v,u) and A+uv :L(u,v)→LA+uv(u,v) on a Banach space B over a field F
A+vu L(v,u) A+uv
L(v) L(u)
-
v u
Fig.2
HilbertFlowSpaceswithDifferentialsoverGraphs 21
with L(v,u) = −L(u,v) and A+vu(−L(v,u)) = −LA+vu(v,u) for ∀(v,u) ∈ E −→G holding with
continuity equation (cid:16) (cid:17)
LA+vu(v,u)=L(v) for v V −→G
∀ ∈
u∈XNG(v) (cid:16) (cid:17)
such as those shown for vertex v in Fig.3 following
L(u1,v) L(v,u4)
L(u1) - - L(u4)
u
1 u
4
A A
1 4
L(u2,v) A2 A5 L(v,u5)
L(u2) - L(v) - L(u5)
u2 A3 v A6 u5
L(u3,v) L(v,u6)
L(u3) - - L(u6)
u3 Fig.3 u6
with a continuity equation
LA1(v,u )+LA2(v,u )+LA3(v,u ) LA4(v,u ) LA5(v,u ) LA6(v,u )=L(v),
1 2 3 4 5 6
− − −
where L(v) is the surplus flow on vertex v.
Particularly, if L(v) = x˙ or constants v ,v V −→G , the continuity flow −→G;L,A
v v
∈
is respectively said to be a complex flow or an action A(cid:16)flow(cid:17), and −→G-flow if A = 1(cid:16)V, wher(cid:17)e
x˙ =dx /dt, x is a variable on vertex v and v is an element in B for v E −→G .
v v v
∀ ∈
(cid:16) (cid:17)
Clearly, an action flow is an equilibrium state of a continuity flow −→G;L,A . We have
shown that Banach or Hilbert space can be extended over topologicalgra(cid:16)phs ([14],(cid:17)[17]), which
can be applied to understanding the reality of things in [15]-[16], and we also shown that
complex flows can be applied to hold on the global stability of biological n-system with n 3
≥
in [19]. For further discussing continuity flows, we need conceptions following.
Definition 1.2 Let B ,B be Banach spaces over a field F with norms and ,
1 2 1 2
k · k k · k
respectively. An operator T:B B is linear if
1 2
→
T(λv +µv )=λT(v )+µT(v )
1 2 1 2
for λ,µ F, and T is said to be continuous at a vector v if there always exist such a number
0
∈
22 LinfanMAO
δ(ε) for ǫ>0 that
∀
T(v) T(v ) <ε
k − 0 k2
if v v <δ(ε) for v,v ,v ,v B .
k − 0k1 ∀ 0 1 2 ∈ 1
Definition 1.3 Let B ,B be Banach spaces over a field F with norms and ,
1 2 1 2
k · k k · k
respectively. An operator T:B B is bounded if there is a constant M >0 such that
1 2
→
T(v)
2
T(v) M v , i.e., k k M
k k2 ≤ k k1 v ≤
1
k k
for v B and furthermore, T is said to be a contractor if
∀ ∈
T(v ) T(v ) c v v )
1 2 1 2
k − k≤ k − k
for v ,v B with c [0,1).
1 2
∀ ∈ ∈
We only discuss the case that all end-operators A+ ,A+ are both linear and continuous.
vu uv
In this case, the result following on linear operators of Banach space is useful.
Theorem1.4 LetB ,B beBanachspacesoverafieldFwithnorms and ,respectively.
1 2 1 2
k·k k·k
Then, a linear operator T:B B is continuous if and only if it is bounded, or equivalently,
1 2
→
T(v)
2
T := sup k k <+ .
k k 06=v∈B1 kvk1 ∞
Let −→G ,→−G , be a graph family. The main purpose of this paper is to extend Ba-
1 2
···
nach or Hnilbert spaceos to Banach or Hilbert continuity flow spaces over topological graphs
−→G ,−→G , andestablishdifferentialsoncontinuityflows,whichenablesonetocharacterize
1 2
···
tnheir globallyochange rate constraint on the combinatorial structure. A few well-known results
such as those of Taylor formula, L’Hospital’s rule on limitation are generalized to continuity
flows, and algebraic or differential flow equations are discussed in this paper. All of these
results form the elementary differential theory on continuity flows, which contributes math-
ematical combinatorics and can be used to characterizing the behavior of complex systems,
particularly, the synchronization.
For terminologies and notations not defined in this paper, we follow references [1] for
mechanics,[4]forfunctionalsandlinearoperators,[22]fortopology,[8]combinatorialgeometry,
[6]-[7],[25] for Smarandache systems, Smarandache geometries and Smaarandache multispaces
and [2], [20] for biological mathematics.
§2. Banach and Hilbert Flow Spaces
2.1 Linear Spaces over Graphs
n
Let −→G ,−→G , ,→−G be oriented graphs embedded in topological space S with −→G = −→G ,
1 2 n i
···
i=1
S
HilbertFlowSpaceswithDifferentialsoverGraphs 23
i.e.,−→G isasubgraphof−→G forintegers1 i n. Inthis case,theseisnaturallyanembedding
i
≤ ≤
ι:−→G −→G.
i
→
Let V be a linear space over a field F. A vector labeling L : −→G V is a mapping with
→
L(v),L(e) V for v V(−→G),e E(−→G). Define
∈ ∀ ∈ ∈
−→GL1 +−→GL2 = −→G −→G L1 −→G −→G L1+L2 −→G −→G L2 (2.1)
1 2 1\ 2 1 2 2\ 1
(cid:16) (cid:17) [(cid:16) \ (cid:17) [(cid:16) (cid:17)
and
λ −→GL =−→Gλ·L (2.2)
·
for λ F. Clearly, if , and −→GL,−→GL1,−→GL2 are continuity flows with linear end-operators
∀ ∈ 1 2
A+ and A+ for (v,u) E −→G , −→GL1 +−→GL2 and λ −→GL are continuity flows also. If we
vu uv ∀ ∈ 1 2 ·
consider each continuity flow(cid:16)−→GL(cid:17)a continuity subflow of −→GL, where L : −→G = L(−→G ) but
i i i
L : −→G −→Gi 0 for integers 1 i n, and define O : −→G b 0, then all continuity flows,
\ → ≤ ≤ →
b
particularly,allcomplexflows,orallactionflowsonorientedgraphs−→G ,−→G , ,→−G naturally
1 2 n
b V ···
formalinearspace,denotedby −→G ,1 i n ;+, overafieldF underoperations(2.1)
i
≤ ≤ ·
and (2.2) because it holds with:(cid:18)D E (cid:19)
(1) A field F of scalars;
V
(2) A set −→G ,1 i n of objects, called continuity flows;
i
≤ ≤
D E
(3) An operation “+”, called continuity flow addition, which associates with each pair of
V V
continuityflows−→GL1,−→GL2 in −→G ,1 i n acontinuityflows−→GL1+−→GL2 in −→G ,1 i n ,
1 2 i ≤ ≤ 1 2 i ≤ ≤
called the sum of −→GL1 and −→GDL2, in such a wEay that D E
1 2
(a) Addition is commutative, −→GL1 +−→GL2 =−→GL2 +−→GL1 because of
1 2 2 1
−→GL1 +−→GL2 = −→G −→G L1 −→G −→G L1+L2 −→G −→G L2
1 2 1− 2 1 2 2− 1
(cid:16) (cid:17)L2[(cid:16) \ (cid:17)L2+L1[(cid:16) (cid:17)L1
= −→G −→G −→G −→G −→G −→G
2 1 1 2 1 2
− −
= −(cid:16)→GL2 +−→GL(cid:17)1; [(cid:16) \ (cid:17) [(cid:16) (cid:17)
2 1
(b) Addition is associative, −→GL1 +−→GL2 +−→GL3 = −→GL1 + −→GL2 +−→GL3 because if we
1 2 3 1 2 3
let (cid:16) (cid:17) (cid:16) (cid:17)
L (x), if x −→G −→G −→G
i i j k
∈ \
 Lj(x), if x −→Gj (cid:16)−→GiS−→Gk(cid:17)
L+ijk(x)= LLL+i+ikkj(((xxx))),,, iiifff xxx∈∈∈∈(cid:16)−→G−−→→GGkii\\T(cid:16)(cid:16)−→−−→→GGGikjS(cid:17)S\\−→G−→−→GGj(cid:17)(cid:17)kj (2.3)
 LLi+j(kx(x))+, Lj(x)+Lk(x) iiff xx∈∈−(cid:16)(cid:16)→G−→GiTjTT−→G−→GjTk(cid:17)(cid:17)−→G\k−→Gi
24 LinfanMAO
and
L (x), if x −→G −→G
i i j
∈ \
L+(x)= L (x), if x −→G −→G (2.4)
ij j j i
 L (x)+L (x), if x∈−→G \ −→G
i j i j
∈
for integers 1 i,j,k n, then T
≤ ≤
L+ L+
−→GL1 +−→GL2 +−→GL3 = −→G −→G 12 +−→GL3 = −→G −→G −→G 123
1 2 3 1 2 3 1 2 3
(cid:16) (cid:17) (cid:16) [ (cid:17) L+ (cid:16) [ [ (cid:17)
= −→GL1 + −→G −→G 23 =−→GL1 + −→GL2 +−→GL3 ;
1 2 3 1 2 3
(cid:16) [ (cid:17) (cid:16) (cid:17)
(c)ThereisauniquecontinuityflowOon−→G holdwithO(v,u)=0for (v,u) E −→G and
∀ ∈
V (cid:16) (cid:17) V
V −→G in −→G ,1 i n ,calledzerosuchthat−→GL+O=−→GL for−→GL −→G ,1 i n ;
i i
≤ ≤ ∈ ≤ ≤
(cid:16) (cid:17) D E V D E
(d) For each continuity flow −→GL −→G ,1 i n there is a unique continuity flow
i
∈ ≤ ≤
−→G L such that −→GL+−→G L =O; D E
− −
(4) An operation “, called scalar multiplication, which associates with each scalar k in F
· V
andacontinuityflow−→GL in −→G ,1 i n acontinuityflowk −→GL inV,calledtheproduct
i
≤ ≤ ·
of k with −→GL, in such a wayDthat E
V
(a) 1 −→GL =−→GL for every −→GL in −→G ,1 i n ;
i
· ≤ ≤
D E
(b) (k k ) −→GL =k (k −→GL);
1 2 1 2
· ·
(c) k (−→GL1 +−→GL2)=k −→GL1 +k −→GL2;
· 1 2 · 1 · 2
(d) (k +k ) −→GL =k −→GL+k −→GL.
1 2 1 2
· · ·
V V
Usually, we abbreviate −→G ,1 i n ;+, to −→G ,1 i n if these operations
i i
≤ ≤ · ≤ ≤
+ and are clear in the cont(cid:18)eDxt. E (cid:19) D E
·
Byoperation(1.1),−→GL1+−→GL2 =−→GL1 ifandonlyif−→G −→G withL :−→G −→G 0and
1 2 6 1 1 6(cid:22) 2 1 1\ 2 6→
−→GL1+−→GL2 =−→GL2 ifandonlyif−→G −→G withL :−→G −→G 0,whichallowsustointroduce
1 2 6 2 2 6(cid:22) 1 2 2\ 1 6→
the conception of linear irreducible. Generally, a continuity flow family −→GL1,−→GL2, ,−→GLn
{ 1 2 ··· n }
is linear irreducible if for any integer i,
−→G −→G with L :−→G −→G 0, (2.5)
i l i i l
6(cid:22) \ 6→
l=i l=i
[6 [6
where 1 i n. We know the following result on linear generated sets.
≤ ≤
Theorem 2.1 Let V be a linear space over a field F and let −→GL1,−→GL2, ,−→GLn be an
1 2 ··· n
linear irreducible family, L : −→G V for integers 1 i nn with linear operatoors A+ ,
i i → ≤ ≤ vu
A+ for (v,u) E −→G . Then, −→GL1,−→GL2, ,−→GLn is an independent generated set of
uv ∀ ∈ 1 2 ··· n
(cid:16) (cid:17) n o
HilbertFlowSpaceswithDifferentialsoverGraphs 25
V
−→G ,1 i n , called basis, i.e.,
i
≤ ≤
D E
V
dim −→G ,1 i n =n.
i
≤ ≤
D E
V
Proof By definition, −→GLi,1 i n is a linear generated of −→G ,1 i n with
i ≤ ≤ i ≤ ≤
L :−→G V, i.e., D E
i i
→ V
dim −→G ,1 i n n.
i
≤ ≤ ≤
D E
We only need to show that −→GLi,1 i n is linear independent, i.e.,
i ≤ ≤
V
dim −→G ,1 i n n,
i
≤ ≤ ≥
D E
which implies that if there are n scalars c ,c , ,c holding with
1 2 n
···
c −→GL1 +c −→GL2 + +c −→GLn =O,
1 1 2 2 ··· n n
thenc =c = =c =0. Notice that −→G ,−→G , ,−→G islinearirreducible. We areeasily
1 2 n 1 2 n
··· { ··· }
know −G→ −→G = and find an element x E(−G→ −→G ) such that c L (x)=0 for integer
i l i l i i
\ 6 ∅ ∈ \
l=i l=i
6 6
i,1 i nS. However,L (x)=0 by (1.5). We get thatSc =0 for integers 1 i n.
i i
≤ ≤ 6 ≤ ≤ 2
V
Asubspaceof −→G ,1 i n iscalledanA -flow spaceifitselementsareallcontinuity
i 0
≤ ≤
flows −→GL with L(vD), v V −→GE are constant v. The result following is an immediately
∈
conclusion of Theorem 2.1. (cid:16) (cid:17)
Theorem 2.2 Let −→G,−→G ,−→G , ,−→G be oriented graphs embedded in a space S and V
1 2 n
···
a linear space over a field F. If −→Gv,−→Gv1,−→Gv2, ,−→Gvn are continuity flows with v(v) =
1 2 ··· n
v,v (v)=v V for v V −→G , 1 i n, then
i i
∈ ∀ ∈ ≤ ≤
(cid:16) (cid:17)
v
(1) −→G is an A -flow space;
0
D E
v v v
(2) −→G11,→−G22,··· ,−→Gnn is an A0-flow space if and only if −→G1 = −→G2 = ··· = −→Gn or
v1 =v2 D= =vn =0. E
···
v v v
Proof By definition, −→G 1 + −→G 2 and λ−→G are A -flows if and only if −→G = −→G or
1 2 0 1 1
v =v =0 by definition. We therefore know this result.
1 2
2
2.2 Commutative Rings over Graphs
Furthermore, if V is a commutative ring (R;+, ), we can extend it overoriented graphfamily
·
−→G ,−→G , ,−→G by introducing operation + with (2.1) and operation following:
1 2 n
{ ··· } ·
−→GL1 −→GL2 = −→G −→G L1 −→G −→G L1·L2 −→G −→G L2, (2.6)
1 · 2 1\ 2 1 2 2\ 1
(cid:16) (cid:17) [(cid:16) \ (cid:17) [(cid:16) (cid:17)
26 LinfanMAO
where L L : x L (x) L (x), and particularly, the scalar product for Rn,n 2 for
1 2 1 2
· → · ≥
x −→G −→G .
1 2
∈
T R
As we shown in Subsection 2.1, −→G ,1 i n ;+ is an Abelian group. We show
i
≤ ≤
R (cid:18)D E (cid:19)
−→G ,1 i n ;+, is a commutative semigroup also.
i
≤ ≤ ·
(cid:18)D E (cid:19)
In fact, define
L (x), if x −→G −→G
i i j
∈ \
L×ij(x)= LLj((xx)), L (x), iiff xx∈−−→→GGj \−→G−→Gi
i j i j
· ∈
 L×T L×
Then,weareeasilyknownthat−→GL1 −→GL2 = −→G −→G 12 = −→G −→G 21 =−→GL2 −→GL1
1 · 2 1 2 1 2 2 · 1
R (cid:16) (cid:17) (cid:16) (cid:17)
for −→GL1,−→GL2 −→G ,1 i n ; by definitionS(2.6), i.e., it is coSmmutative.
∀ 1 2 ∈ i ≤ ≤ ·
(cid:18)D E (cid:19)
Let
L (x), if x −→G −→G −→G
i i j k
∈ \
 Lj(x), if x −→Gj (cid:16)−→GiS−→Gk(cid:17)
L×ijk(x)= LLLiikkj(((xxx))),,, iiifff xxx∈∈∈(cid:16)−→G−−→→GGkii\\T(cid:16)(cid:16)−→−−→→GGGikjS(cid:17)S\−→G−→−→GGj(cid:17)(cid:17)kj
∈ \
Then,  LLij(kx(x))·,Lj(x)·Lk(x) iiff xx∈∈−(cid:16)(cid:16)→G−→GiTjTT−→G−→GjTk(cid:17)(cid:17)−→G\k−→Gi
L× L×
−→GL1 −→GL2 −→GL3 = −→G −→G 12 −→GL3 = −→G −→G −→G 123
1 · 2 · 3 1 2 · 3 1 2 3
(cid:16) (cid:17) (cid:16) [ (cid:17) L× (cid:16) [ [ (cid:17)
= −→GL1 −→G −→G 23 =−→GL1 −→GL2 −→GL3 .
1 · 2 3 1 · 2 · 3
(cid:16) [ (cid:17) (cid:16) (cid:17)
Thus,
−→GL1 −→GL2 −→GL3 =−→GL1 −→GL2 −→GL3
1 · 2 · 3 1 · 2 · 3
(cid:16) (cid:17) (cid:16) (cid:17)
R
for −→GL,−→GL1,−→GL2 −→G ,1 i n ; , which implies that it is a semigroup.
∀ 1 2 ∈ i ≤ ≤ ·
(cid:18)D E (cid:19)
We are also need to verify the distributive laws, i.e.,
−→GL3 −→GL1 +−→GL2 =−→GL3 −→GL1 +−→GL3 −→GL2 (2.7)
3 · 1 2 3 · 1 3 · 2
(cid:16) (cid:17)
and
−→GL1 +−→GL2 −→GL3 =−→GL1 −→GL3 +−→GL2 −→GL3 (2.8)
1 2 · 3 1 · 3 2 · 3
(cid:16) (cid:17)
HilbertFlowSpaceswithDifferentialsoverGraphs 27
R
for −→G ,−→G ,−→G −→G ,1 i n ;+, . Clearly,
3 1 2 i
∀ ∈ ≤ ≤ ·
(cid:18)D E (cid:19)
L+ L×
−→GL3 −→GL1 +−→GL2 = −→GL3 −→G −→G 12 = −→G −→G −→G 3(21)
3 · 1 2 3 · 1 2 3 1 2
(cid:16) (cid:17) (cid:16) [L× (cid:17) (cid:16) (cid:16)L× [ (cid:17)(cid:17)
= −→G −→G 31 −→G −→G 32 =−→GL3 −→GL1 +−→GL3 −→GL2,
3 1 3 2 3 · 1 3 · 2
(cid:16) [ (cid:17) [(cid:16) [ (cid:17)
which is the (2.7). The proof for (2.8) is similar. Thus, we get the following result.
Theorem 2.3 Let (R;+, ) be a commutative ring and let −→GL1,−→GL2, ,−→GLn be a linear
· 1 2 ··· n
irreducible family, L : −→G R for integers 1 i n wnith linear operators Ao+ , A+ for
i i → R ≤ ≤ vu uv
(v,u) E −→G . Then, −→G ,1 i n ;+, is a commutative ring.
i
∀ ∈ ≤ ≤ ·
(cid:16) (cid:17) (cid:18)D E (cid:19)
2.3 Banach or Hilbert Flow Spaces
V
Let {−→GL11,−→GL22,··· ,−→GLnn} be a basis of −→Gi,1≤i≤n , where V is a Banach space with a
DV E
norm . For −→GL −→G ,1 i n , define
i
k · k ∀ ∈ ≤ ≤
D E
−→GL = L(e) . (2.9)
k k
(cid:13) (cid:13) e EX→−G
(cid:13) (cid:13) ∈
(cid:13) (cid:13) (cid:16) (cid:17)
V
Then, for −→G,−→GL1,−→GL2 −→G ,1 i n we are easily know that
∀ 1 2 ∈ i ≤ ≤
D E
(1) −→GL 0 and −→GL =0 if and only if −→GL =O;
≥
(cid:13) (cid:13) (cid:13) (cid:13)
(2) (cid:13)(cid:13)−→GξL(cid:13)(cid:13) =ξ −→GL(cid:13)(cid:13) for(cid:13)(cid:13)any scalar ξ;
(cid:13) (cid:13) (cid:13) (cid:13)
(3) (cid:13)(cid:13)−→GL11(cid:13)(cid:13)+−→GL2(cid:13)(cid:13)2 ≤(cid:13)(cid:13) −→GL11 + −→GL22 because of
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
−→GL1 +(cid:13)−→GL2 = (cid:13) (cid:13) (cid:13) L(cid:13) (e)(cid:13)
1 2 k 1 k
(cid:13)(cid:13) (cid:13)(cid:13) e∈E X→−G1\→−G2
(cid:13) (cid:13) (cid:16) (cid:17)
+ L (e)+L (e) + L (e)
1 2 2
k k k k
e∈E →−GX1 →−G2 e∈E X→−G2\→−G1
(cid:16) T (cid:17) (cid:16) (cid:17)
L (e) + L (e)
 1 1 
≤ k k k k
e∈E X→−G1\→−G2 e∈E →−GX1 →−G2 
 (cid:16) (cid:17) (cid:16) T (cid:17) 
+ L (e) + L (e) = −→GL1 + −→GL2 .
 k 2 k k 2 k 1 2
e∈E X→−G2\→−G1 e∈E →−GX1 →−G2  (cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13)
 (cid:16) (cid:17) (cid:16) T (cid:17)  (cid:13) (cid:13) (cid:13) (cid:13)
for L (e)+L (e) L (e) + L (e) in Banach space V. Therefore, is also a norm
1 2 1 2
k k ≤ k k k k k · k
28 LinfanMAO
V
on −→G ,1 i n .
i
≤ ≤
D E
V
Furthermore,ifV isaHilbertspacewithaninnerproduct , ,for −→GL1,−→GL2 −→G ,1 i n ,
h· ·i ∀ 1 2 ∈ i ≤ ≤
define D E
−→GL1,−→GL2 = L (e),L (e)
1 2 h 1 1 i
D E e∈E X→−G1\→−G2
(cid:16) (cid:17)
+ L (e),L (e) + L (e),L (e) . (2.10)
1 2 2 2
h i h i
e∈E →−GX1 →−G2 e∈E X→−G2\→−G1
(cid:16) T (cid:17) (cid:16) (cid:17)
Then we are easily know also that
V
(1) For −→GL −→G ,1 i n ,
i
∀ ∈ ≤ ≤
D E
−→GL,−→GL = L(e),L(e) 0
h i≥
D E e EX→−G
∈
(cid:16) (cid:17)
and −→GL,−→GL =0 if and only if −→GL =O.
D E
V
(2) For −→GL1,−→GL2 −→Gi,1 i n ,
∀ ∈ ≤ ≤
D E
−→GL1,−→GL2 = −→GL2,−→GL1
1 2 2 1
D E D E
because of
−→GL1,−→GL2 = L (e),L (e) + L (e),L (e)
1 2 h 1 1 i h 1 2 i
D E e∈E X→−G1\→−G2 e∈E →−GX1 →−G2
(cid:16) (cid:17) (cid:16) T (cid:17)
+ L (e),L (e)
2 2
h i
e∈E X→−G2\→−G1
(cid:16) (cid:17)
= L (e),L (e) + L (e),L (e)
1 1 2 1
h i h i
e∈E X→−G1\→−G2 e∈E →−GX1 →−G2
(cid:16) (cid:17) (cid:16) T (cid:17)
+ L (e),L (e) = −→GL2,−→GL1
h 2 2 i 2 1
e∈E X→−G2\→−G1 D E
(cid:16) (cid:17)
for L (e),L (e) = L (e),L (e) in Hilbert space V.
1 2 2 1
h i h i
V
(3) For −→GL,→−GL1,→−GL2 −→G ,1 i n and λ,µ F, there is
1 2 ∈ i ≤ ≤ ∈
D E
λ−→GL1 +µ−→GL2,→−GL =λ −→GL1,−→GL +µ −→GL2,−→GL
1 2 1 2
D E D E D E