This short paper would finally write on signal through the relation between language and nerve, for which using mathematical method at the way.
This paper's one of kernels is energy which is naturally accepted at the side of verve, but at the side of language, it may be not accepted widely till now.
In this paper, I probably do not refer to the language's energy, that has been written several times in the papers before.
Around 2008, I was thinking on energy in language related with distance in language. Distance is one of the kernel themes of my study of language universals in those days. In simply saying, distance is the result of movement and for movement there must inevitably needs energy. So I had thought that if language have distance, there must be energy or its alike in language that is supposed in mathematical models. But in my ability in 2008, I could not develop the deep and wide range of language from the theme, energy and distance. So Energy Distance Theory was still now unfinished.
0 The world spreads around the human being.
1 Language divides the world.
2 Language connects the world.
3 Language bends the world.
4 Language stretches the world.
5 Language shrinks the world.
6 Language extinguishes the world.
7 Language creates the world.
8 Language gives despair.
9 Language gives hope.
10 Language is pasting on spacetime with energy.
On language and human sense, I ever wrote simply at the next paper.
Macro Time and Micro time
TANAKA Akio
24 July 2013
atbankofdam
1. Through natural language, in human being, occurred the electrical signal by eye or ear. These complex situations are beyond this paper’s limits.
2. Language is a physical object as signal and its transmission. At this circumstances, language must be recognised to be the existence that has finite time.
3. An apple on the desk gradually becomes rotten by passing the time very after the crop in the orchard. #0
4. Like an apple, language has passing physical time in oneself.
5. Language is metamorphosed by the time progressing. #1
6. Language includes the outer world from human being to universe. At this declaration, I recall Blaise Pascal’s Pensées. XXXIII. PROOFS OF JESUS CHRIST 308 The infinite distance between body and mind symbolizes the infinitely more infinite distance between mind and charity, for charity is supernatural.(Translated by A.J. Krailsheimer, 1966) #2
7. Language’s time goes freely from the present to the future or the present to the past. #3
8. Language symbolises the time from finiteness to infinity. #4
9. Human being recognises this vast language world perfectly. #5
References
#0 For WITTGENSTEIN Ludwig Position of Language / December 10, 2005 – August 3, 2012 / Sekinan Research Field of Language
#1 Time of Word / Complex Manifold Deformation Theory / January 1, 2009 / sekinanlogos
#2 PASCAL PENSÉES. Translated with an introduction by A.J. Krailsheimer. PENGUIN BOOKS 1966.
#3 Escalator language and Time For SHINRAN’s Idea and BOHDISATTVA / Escalator Language Theory / December 16, 2006 / Sekinan Research Field of Language
#4 From Finiteness to Infinity on Language / Topological Group Theory / February 1, 2009 / sekinanlogos
#5 Understandability of Language / Complex Manifold Deformation Theory /January 9, 2009 / sekinanlogos
At this chapter, the most important concept is positive quantum and negative quantum.
Negative quantum changed to positive quantum by energy.
At the linear space, quantum changes positive to negative and also negative to positive.
By this model, on-off phenomena are mapped at the linear space's quanta's situation.
In the models, quantum is led from V.G.Drinfel'd and M.Jinbo's Quantum group originated in 1985.
Reference
JIMBO Michio. Quantum group and Yang-Baxter equation. Maruzen Shuppan. Tokyo. 2012
5.
Method
Manuscript of Quantum Theory for Language written in spring 2003 was roughly designed at a days I was in
hospital by pneumonia half a month in October 2002, when I always saw the river and the mountains west end of Tokyo. I was the very reviewing life time for my research work.
My poor study was restricted in a narrow field of Chinese classical linguistics mainly developed in the late Qing dynasty the latter half of the 19th century,represented by DUAN Yucai, WANG Niansun, WANG Yingzhi, my favourite WANG Guowei and so forth.
So I determined that my approach to language was only in it and it was the most intimate for me at that time and probably herein after considering my tiny accumulation of study.
1970s' dream, writing clear description on language universals by mathematics
From The Days of Ideogram
4. Time property in characters
In Autumn 2002 I got pneumonia and was hospitalized about 2 weeks, where I thought of 1970s' dream, writing clear description on language universals by mathematics. The theme was as hard as ever. So, at the bed I thought the basis of language from the side of Chinese character’s classical approach which had vast heritage till Qing dynasty. I directed my attention to the character's figure which had compound meanings containing time elements continuing from Yin dynasty's hieroglyphic characters left on bones and tortoise carapaces some 2400 years ago. I thought that Chinese characters had containing time and its structure could be written by geometric approach once I had abandoned for difficulty. After leaving hospital, I wrote a paper titled On Time Property Inherent in Characters*3-1.
2.
Word Indexed is the word that has index in a word, by which meanings are separated in a word at waiting situation. In a word meanings are combined by algebraic axioms and theorems starting from group theory.
Word Indexed seems to be more simply structure than Word Synthesized.
Word indexed basis is at the below.
In Chinese one character has one meaning and becomes a word in classical usage. Modern Chinese has many words that contain two or over two characters for one meaning, but basically almost all the characters have still classical one-character-one-meaning usage at the root of language.
Shuowenjezi Zhu written by DUAN Yucai typically shows some 2,000 year history of characters and their meanings.
Modern Chinese precisely said by Hanyu is one language of the over 50 languages officially recognized at the research, and I only know several language's grammars by the field work by linguists.
Xixiayu is a very interesting for me resembling the Hanyu and in the late 20th century the language was deciphered by Japanese linguist NISHIDA Tatsuo.
On 20 July 2016 I went Tokyo National Museum, Ueno Park, Tokyo to see the exhibition A JOURNEY TO THE IMMORTALS: TREASURES OF ANCIENT GREECE, where I saw the linear A and B. It reminds me the youth days, so to say, the days of decipherment.
1960s -1970s is the age of decipherment in a sense. I was age 20 in 1967 and was learning language and literature at university. In 1958 John Chadwick's THE DECIPHERMENT OF LINEAR B was published from Cambridge University Press. At the preface of the book he wrote that the decipherment of linear B was told at Documents in Mycenaean Greek (Cambridge University Press, 1956) and Michael Ventris that deciphered the Linear B.
In the same age in Japan, Xixia wenzi (Xixia characters) in China was deciphered by NISHIDA Tatsuo (1928-2012) who wrote the analysis and grammar of Xixia characters through the paper Seikamoji no bunseki narabini Seikago bunpou no kenkyuu in 1962.
In almost the same time, Inca characters were studying to decipher. I frequently heard that Russian team developed largely.
In early 1970s I frequently went to Kanda, Tokyo where old bookshops were selling vast Oriental books at the Hakusan street and Yasukuni Street. I bought Chinese classics, especially linguistic classics written in the Qing dynasty and I read them almost every day containing the comparison with the western linguistic results. The Qing dynasty's heritage were DUAN Yucai, WANG Niansun, WANG Yingzhi and WANG Guowei and so forth. DUAN Yucai's Showenjezi zhu and WANG Guowei's Guantang jilin were the most important for me.
In France, 1960s was the days of Bourbaki that was one of the decipher of geometry by algebra, at least I thought so at that time. I sought and bought several Bourbaki's books at the old bookshops in Kanda, Tokyo,which is the largest old bookshop streets in Japan. But from my ability to mathematics Bourbaki was too much difficult to read on. From the days the long and winding road began to mathematics and its applicable study for language universals.
At the exhibition of ancient Greece I confirmed in particular that the stability of language was kept by letters and characters from the Linear A and Linear B.
These language or character's situation especially of ideogram has become my study's foundation.
1 Quantum of language is the smallest unit of language.
2 Quantum of language moves linearly on the floor of language.
3 Linear movement is the properties of quantum.
4 Floor of language is on the space of language.
5 The space of language is two dimensions.
6 Two dimensions are horizontal and vertical.
7 Horizontal movement makes word, #1
8 Vertical movement makes sentence. #1
9 The space of language is electrical digitized place.
10 Chinese /jiao shi/ means classroom in English.
11 /Jiao/ is a quantum of language.
12 /Shi/ is a quantum of language.
13 /Jiao shi/ is a word.
14 /Jiao/ sends a quantum to /shi/ quantum.
15 /Shi/ quantum receives a quantum from /jiao/ quantum.
16 What sends quantum is called positive.
17 What receives quantum is called negative.
18 Quantum has positive energy in original condition.
19 Quantum changes negative in the situation of quanta set.
20 Quantum change occurs in two situations in general.
21 One situation is what quanta stand side by side on a floor and neighboring quanta connect well. #2
22 The other situation is what quanta change oneself by the non-use of quanta meaning in language history progress. #2
23 Word has a positive- negative construction.
24 Positive-negative construction occurs on a floor.
25 Sentence has a positive-positive construction.
26 Positive-positive construction occurs on different floors.
27 The latter quantum transfers on a different floor. This transfer is called .†
28 Quantum has electrical energy which flows to the electrical zero level.
29 Electrical zero level is a sentence end where quantum of language ideally accord with the real world. #3
30 A floor of language is a non-branches electrical circuit.
31 Word is a non-branch circuit.
32 Sentence is a branch circuit.
33 The meaning of word and sentence is a compound system of electrical signals.
#1 Definition of word and sentence can be seen in the paper “Method of Linguistics” and other papers on the site of Sekinan Research Field of Language /www.sekinan.org/.
At the end of paper 2017, I wrote as the following, where I showed the early intuitive papers related with symmetry or mirror. This concept has succeeded till now and a little developed a new direction towards mathematical based concept especially of quantum group.
The concept called symmetry is very important to describe the complex situation of natural language.
Symmetry contains undifferentiated factors in itself, for example mirror, distance,ant-world and so forth.
I ever tried to cultivate this fantastic field to resolve the hardship on language universals one more step up.
My trying paper is the following.
Language based on quantum emerged from thinking the simplest model for containing the finite essential elements of language in summer 2003 at Hakuba, Nagano, Japan at the skirts of Japan Alps. Details are the next.
In August 2003, I went to Hakuba in Nagano prefecture for the summer vacation with my family. At that time I had been thinking on the form of language for which I wrote the paper, that connects with time inherent in characters, in March 2003 also at Hakuba.
At night of August 23 in cottage, I casually saw the advertising paper of electric dictionary. The paper was brought from the convenience store near the cottage in the evening. The dictionary on the paper was Seiko’s English-Japanese dictionary that has additionally consultation for Chinese or French language with large scale. I vaguely considered that after this dictionaries are necessarily taken these multi-lingual way.
At the time I suddenly realized that the form of language may be spherical style in which language contains all the information in itself.That was rather satisfied solution for the tough problem of language that I had been carrying in my life from my twenties.
I wrote the sketch-like paper of the theoretical approach after returning home of Tokyo. The paper was read at the international symposium of UNESCO opened in winter 2003 at Nara. In the paper, the spherical substance of language is seemed to be quantum in DELBRUCK’s image-like physical world. After 5 years from the inspiration at summer of Hakuba, now I consider that spherical essence is manifold in infinite dimensional world.
Now I also realize that the toughest problem of language is minutely solvable in mathematical approach that has structurally definable terms.
In 2006 I wrote Escalator Language series.
Paper, Turning Point of Time is a intuitive paper for the three papers of Distance Theory Algebraically Supplemented 2007.
Distance is one of the most important elements of my language model of language universals. For this element algebraic approach seems to be clearer description to the model. Refer to the next paper.
1
Distance theory is an extension of Quantum Theory for Language.
2
Distance theory is an extension of strength rule in Quantum Theory for Language.
3
Distance theory is considered for the purpose of the guarantee to language.
4
What quanta of language propel to the end of the sentence is for the purpose of the guarantee to language, in which quanta of language finally unite the real world in the end of propelling.
5
The guarantee to the inherent signification of indicator in quantum of language is quantified by the distance which starts from the real world to the quantum of language.
6
A quantum consists of indicators.
An indicator has a signification and a period inherently.
The structure of quantum is indicated in Quantum Theory for Language.
7
An inherent signification is an element in a quantum.
An inherent time is an element in a quantum.
There are two types of elements, significant and periodical.
Element is defined.
8
A significant element gets a signification from the real world.
A periodical element gets a time from the real world.
9
An indicator gets a meaning and a period from elements.
10
An element emerges from the real world to the language world.
An indicator gets power from the elements in the language world.
A quantum moves in the language world by the power of indicators.
11
An element emerges to the language world, because each element has immanent perceptible area which works upon visual sensation and auditory sensation of the human beings.
12
An indicator gets energy in the language world, because each indicator has a tendency which will approach and finally coincide with the real world.
This continuous tendency guarantees the trust in language for the human beings.
13
A quantum moves in the language world toward the real world.
A quantum is not guaranteed in the situation of cessation.
A quantum is guaranteed by the connection to the real world.
Therefore a quantum propels to the real world.
14
Indicators make meaning and connection rule in a quantum, both are derived from significant and periodical elements in an indicator.
15
Meaning is guaranteed by the tendency of coincidence with the real world.
Guarantee of the meaning is reduced by the remoteness of distance from the real world.
16
Connection rule is decided by periodical elements in indicators.
Details are indicated in Quantum Theory for Language.
17
Signification in an indicator and meaning in a quantum once emerged are occasionally transformed or expanded in the language world.
This alteration is called multiplication.
Multiplication is defined.
18
Multiplication generally occurs by the addition of signification in an indicator.
But multiplication in meaning of a quantum sometimes occurs without any addition oneself.
19
Multiplication in a quantum without addition occurs by situational transition in the language world.
20
Situational transition in a quantum is caused by difference of distance from the real world.
Difference of distance at a quantum is a proceeding of abstract thinking in human beings.
21
A quantum of language itself becomes in the language world. Word is defined. Therefore each word has a distance toward the real world. A distance immanent in a word does not emerge itself. Distance emerges in the linear situation of words gathering. This situation is called . Sentence is defined. Therefore sentence is an emergence of distance in words gathering. Words form a line, thereafter one arrangement is determined. Sentence is realized in our world. 22 In Chinese language, /lai/ come has a larger distance than /liao/completion. Words are arranged from the end of a sentence, according to the own- possessing- distance. Therefore /lai le/ having come is realized.
On symmetry I wrote Symmetry Flow Language and Symmetry Flow language 2 in 2007.
Contents are the next.
Symmetry Flow Language
On Symmetry of Language and Time
1 Premise for Symmetry Flow in Language 2 Riemannian Metric, Flow and Entropy 3 Leaf of Language Pourparlers> 4 Homology on Language 5 Simplex, Simplicial Complex and Polyhedron 6 Meaning Variation and Time Shift in Word As Homotopy
Symmetry Flow Language2
On Symmetry of Language and Time 2
1 Boundary, Deformation and Torus as Language 2 Time Shift of Meaning in Moduli Space
2 Space that is deformed successively is described by parameter that is called moduli.
3 Set M consists of all of moduli.
4 Moduli space M (M) is presented for variation of word’s meaning.
5 Parameter t is presented for time shift of meaning in word.
6 Calabi-Yau manifold Khas two moduli that are deformation of complex structure and Kähler manifold. Moduli have symmetry that is called Mirror symmetry.
7 K’s Ricci tensor is the following.
Rij¯ = 0 i is regular coordinate. j¯ is non-regular coordinate.
At language universals, distance and time are the kernel concepts for my language models.
Measure of distance is deeply related with time which is transcendental unit still now.
Category theory is probably very useful for the relation between distance and time.
For these concepts, I ever wrote several trial papers. Refer to the next one.
Derived category is abelian category’s coherent sheaf’s complex that is composition of successive arrows becomes 0.
7 From derived category, distinguished triangle is presented.
8 Here time conjecture of language is presented.
(1)Distinguished triangle makes the model for shift of time on language.
(2)Time on language is closed, successive and circular in word.
Circulation is worked between starting point and ending point of word.
Paper group, Language and Spacetime is especially focused at time in space.
Every paper of Language and Spacetime, Symmetry Flow Language and Symmetry Flow Language 2
can be seen at the at the site, SRFL Paper Top pager's right column's Paper 2003-2007.
Signal generates language as the Morse code generates letters and language.
But language does not generate signal code because language has not electric energy.
It maybe that signal is the root of language.Truly or not?
What is signal?
What is generation?
I once wrote a trial paper, Generation Theorem in 2008. Text is the below.
von Neumann Algebra 2
Note
Generation Theorem
TANAKA Akio
[Main Theorem]
<Generation theorem>
Commutative von Neumann Algebra N is generated by only one self-adjoint operator.
[Proof outline]
N is generated by countable {An}.
An = *An
Spectrum deconstruction An = ∫1-1λdEλ(n)
C*algebra that is generated by set { Eλ(n) ; λ∈Q∩[-1, 1], n∈N} A
A’’ = N
A is commutative.
I∈A
Existence of compact Hausdorff space Ω = Sp(A )
A = C(Ω)
Element corresponded with f∈C(Ω) A∈A
N is generated by A.
[Index of Terms]
|A|Ⅲ7-5
|| . ||Ⅱ2-2
||x||Ⅱ2-2
<x, y>Ⅱ2-1
*algebraⅡ3-4
*homomorphismⅡ3-4
*isomorphismⅡ3-4
*subalgebraⅡ3-4
adjoint spaceⅠ12
algebraⅠ8
axiom of infinityⅠ1-8
axiom of power setⅠ1-4
axiom of regularityⅠ1-10
axiom of separationⅠ1-6
axiom of sumⅠ1-5
B ( H )Ⅱ3-3
Banach algebraⅡ2-6
Banach spaceⅡ2-3
Banach* algebraⅡ2-6
Banach-Alaoglu theoremⅡ5
basis of neighbor hoodsⅠ4
bicommutantⅡ6-2
bijectiveⅡ7-1
binary relationⅡ7-2
boundedⅡ3-3
bounded linear operatorⅡ3-3
bounded linear operator, B ( H )Ⅱ3-3
C* algebraⅡ2-8
cardinal numberⅡ7-3
cardinality, |A|Ⅱ7-5
characterⅡ3-6
character space (spectrum space), Sp( )Ⅱ3-6
closed setⅠ2-2
commutantⅡ6-2
compactⅠ3-2
complementⅠ1-3
completeⅡ2-3
countable setⅡ7-6
countable infinite setⅡ7-6
coveringⅠ3-1
commutantⅡ6-2
D ( )Ⅱ3-2
denseⅠ9
dom( )Ⅱ3-2
domain, D ( ), dom( )Ⅱ3-2
empty setⅠ1-9
equal distance operatorⅡ4-1
equipotentⅢ7-1
faithfulⅡ3-4
Gerfand representationⅡ3-7
Gerfand-Naimark theoremⅡ4
HⅡ3-1
Hausdorff spaceⅠ5
Hilbert spaceⅡ3-1
homomorphismⅡ3-4
idempotent elementⅡ9-1
identity elementⅡ9-1
identity operatorⅡ6-1
injectiveⅢ7-1
inner productⅡ2-1
inner spaceⅠ6
involution*Ⅰ10
linear functionalⅡ5-2
linear operatorⅡ3-2
linear spaceⅠ6
linear topological spaceⅠ11
locally compactⅠ3-2
locally vertexⅠ11
NⅢ3-8
N1Ⅲ3-8
neighborhoodⅠ4
normⅡ2-2
normⅡ3-3
norm algebraⅡ5
norm spaceⅡ2-2
normalⅡ2-4
normalⅡ3-4
open coveringⅠ3-2
open setⅠ2-2
operatorⅡ3-2
ordinal numberⅡ7-3
productⅠ8
product setⅡ7-2
r( )Ⅱ2
R ( )Ⅱ3-2
ran( )Ⅱ3-2
range, R ( ), ran( )Ⅱ3-2
reflectiveⅠ12
relationⅢ7-2
representationⅡ3-5
ringⅠ7
Schwarz’s inequalityⅡ2-2
self-adjointⅡ3-4
separableⅡ7-7
setⅠ7
spectrum radius r( )Ⅱ2
Stone-Weierstrass theoremⅡ1
subalgebraⅠ8
subcoveringⅠ3-1
subringⅠ7
subsetⅠ1-3
subspaceⅠ2-3
subtopological spaceⅠ2-3
surjectiveⅢ7-1
system of neighborhoodsⅠ4
τs topologyⅡ7-9
τw topologyⅡ7-9
the second adjoint spaceⅠ12
topological spaceⅠ2-2
topologyⅠ2-1
total order in strict senseⅡ7-3
ultra-weak topologyⅢ6-4
unit sphereⅡ5-1
unitaryⅡ3-4
vertex setⅡ3-3
von Neumann algebraⅡ6-3
weak topologyⅡ5-3
weak * topologyⅡ5-3
zero elementⅡ9-1
[Explanation of indispensable theorems for main theorem]
ⅠPreparation
<0 Formula>
0-1 Quantifier
(i) Logic quantifier ┐ ⋀⋁ → ∀∃
(ii) Equality quantifier =
(iii) Variant term quantifier
(iiii) Bracket [ ]
(v) Constant term quantifier
(vi) Functional quantifier
(vii) Predicate quantifier
(viii) Bracket ( )
(viiii) Comma ,
0-2 Term defined by induction
0-3 Formula defined by induction
<1 Set>
1-1 Axiom of extensionality ∀x∀y[∀z∈x↔z∈y]→x=y.
1-2 Seta, b
1-3 a is subset of b. ∀x[x∈a→x∈b].Notation is a⊂b. b-a = {x∈b ; x∉a} is complement of a.
1-4 Axiom ofpower set∀x∃y∀z[z∈y↔z⊂x]. Notation is P (a).
1-5 Axiom of sum ∀x∃y∀z[z∈y↔∃w[z∈w∧w∈x]]. Notation is ∪a.
1-6 Axiom of separationx, t= (t1, …, tn), formula φ(x, t) ∀x∀t∃y∀z[z∈y↔z∈x∧φ(x, t)].
1-7 Proposition of intersection {x∈a ; x∈b} = {x∈b; x∈a} is set by axiom of separation. Notation is a∩b.
1-8 Axiom of infinity∃x[0∈x∧∀y[y∈x→y∪{y}∈x]].
1-9 Proposition of empty set Existence of set a is permitted by axiom of infinity. {x∈a; x≠x} is set and has not element. Notation of empty set is 0 or Ø.
1-10 Axiom of regularity ∀x[x≠0→∃y[y∈x∧y∩x=0].
<2 Topology>
2-1
Set X
Subset of power set P(X) T
T that satisfies next conditions is called topology.
(i) Family of X’s subset that is not empty set <Ai;i∈I>, Ai∈T→∪i∈I Ai is belonged to T.
(ii) A, B∈T→A∩B∈T
(iii) Ø∈T, X∈T.
2-2
Set having T, (X, T),is called topological space, abbreviated to X, being logically not confused.
Element of T is called open set.
Complement of Element of T is called closed set.
2-3
Topological space (X, T)
Subset of XY
S ={A∩Y ; A∈T}
Subtopological space (Y, S)
Topological space is abbreviated to subspace.
<3 Compact>
3-1
Set X
Subset of XY
Family of X’s subset that is not empty set U = <Ui; i∈I>
U is covering of Y. ∪U = ∪i∈I⊃Y
Subfamily of UV = <Ui; i∈J > (J⊂I)
V is subcoveringof U.
3-2
Topological space X
Elements of U Open set of X
U is called open covering of Y.
When finite subcovering is selected from arbitrary open covering of X, X is called compact.
When topological space has neighborhood that is compact at arbitrary point, it is called locally compact.
<4 Neighborhood>
Topological space X
Point of Xa
Subset of X A
Open set B
a∈B⊂A
A is called neighborhood of a.
All of point a’s neighborhoods is called system of neighborhoods.
System of neighborhoods of point aV(a)
Subset of V(a) U
Element of U B
Arbitrary element of V(a) A
When B⊂A, U is called basis of neighborhoods of point a.
<5 Hausdorff space>
Topological space X that satisfies next condition is called Hausdorff space.
Distinct points of X a, b
Neighborhood of aU
Neighborhood of bV
U∩V = Ø
<6 Linear space>
Compact Hausdorff space Ω
Linear space that is consisted of all complex valued continuous functions over Ω C(Ω)
When Ω is locally compact, all complex valued continuous functions over Ω, that is 0 at infinite point is expressed by C0(Ω).
<7 Ring>
Set R
When R is module on addition and has associative law and distributive law on product, R is called ring.
When ring in which subset S is not φ satisfies next condition, S is called subring.
a, b∈S
ab∈S
<8 Algebra>
C(Ω) and C0(Ω) satisfy the condition of algebra at product between points.
Subspace A ⊂C(Ω) or A⊂C0(Ω)
When A is subring, A is called subalgebra.
<9 Dense>
Topological space X
Subset of X Y
Arbitrary open set that is not Ø in X A
When A∩Y≠Ø, Y is dense in X.
<10 Involution>
Involution* over algebra A over C is map * that satisfies next condition.
Map * : A∈A↦A*∈A
Arbitrary A, B∈A, λ∈C
(i) (A*)* = A
(ii) (A+B)* = A*+B*
(iii) (λA)* =λ-A*
(iiii) (AB)* = B*A*
<11 Linear topological space>
Number field K
Linear space over KX
When X satisfies next condition, X is called linear topological space.
(i) X is topological space
(ii) Next maps are continuous.
(x, y)∈X×X↦x+y∈X
(λ, x)∈K×X↦λx∈X
Basis of neighborhoods of X’ zero element 0 V
When V⊂V is vertex set, X is called locally vertex.
<12 Adjoint space>
Norm space X
Distance d(x, y) = ||x-y|| (x, y∈X )
X is locally vertex linear topological space.
All of bounded linear functional over X X*
Norm of f∈X* ||f||
X* is Banach space and is called adjoint space of X.
Adjoint space of X* is Banach space and is called the second adjoint space.
When X = X*, X is called reflective.
ⅡIndispensable theorems for proof
<1Stone-Weierstrass Theorem>
Compact Hausdorff space Ω
Subalgebra A ⊂C(Ω)
When A ⊂C(Ω) satisfies next condition, A is dense at C(Ω).
(i) A separates points of Ω.
(ii) f∈A →f－∈A
(iii) 1∈A
Locally compact Hausdorff space Ω
Subalgebra A⊂C0(Ω)
When A⊂C0(Ω) satisfies next condition, A is dense at C0(Ω).
(i) A separates points of Ω.
(ii) f∈A → f－∈A
(iii) Arbitrary ω∈A , f∈A , f(ω) ≠0
<2Norm algebra>
C* algebra A
Arbitrary element of A A
When A is normal, limn→∞||An||1/n= ||A||
limn→∞||An||1/nis called spectrum radius of A. Notation is r(A).
[Note for norm algebra]
<2-1>
Number field K = R or C
Linear space over KX
Arbitrary elements of Xx, y
< x, y>∈K satisfies next 3 conditions is called inner product of x and y.
Arbitrary x, y, z∈X,λ∈K
(i) <x, x> ≧0, <x, x> = 0 ⇔x = 0
(ii) <x, y> =
(iii) <x,λy+z> = λ<x, y> + <x, z>
Linear space that has inner product is called inner space.
<2-2>
||x|| = <x, x>1/2
Schwarz’s inequality
Inner space X
|<x, y>|≦||x|| + ||y||
Equality consists of what x and y are linearly dependent.
||・|| defines norm over X by Schwarz’s inequality.
Linear space that has norm || ・|| is called norm space.
<2-3>
Norm space that satisfies next condition is called complete.
un∈X (n = 1, 2,…), limn, m→∞||un– um|| = 0
u∈X limn→∞||un– u|| = 0
Complete norm space is called Banach space.
<2-4>
Topological space X that is Hausdorff space satisfies next condition is called normal.
Closed set of XF, G
Open set of XU, V
F⊂U, G⊂V, U∩V = Ø
<2-5>
When A satisfies next condition, A is norm algebra.
A is norm space.
∀A, B∈A
||AB||≦||A|| ||B||
<2-6>
When A is complete norm algebra on || ・ ||, A is Banach algebra.
<2-7>
When A is Banach algebra that has involution * and || A*|| = ||A|| (∀A∈A), A is Banach * algebra.
<2-8>
When A is Banach * algebraand ||A*A|| = ||A||2(∀A∈A) , A is C*algebra.
<3 Commutative Banach algebra>
Commutative Banach algebra A
Arbitrary A∈A
Character X
|X(A)|≦r(A)≦||A||
[Note for commutative Banach algebra] ( ) is referential section on this paper.
<3-1 Hilbert space>
Hilbert space inner space that is complete on norm ||x|| Notation is H.
<3-2 Linear operator>
Norm space V
Subset of VD
Element of Dx
Map T : x→Tx∈V
The map is called operator.
D is called domain of T. Notation is D( T ) or dom T.
Set A⊂D
Set TA {Tx : x∈A}
TD is called range of T. Notation is R (T) or ran T.
α , β∈C, x, y∈D ( T )
T(αx+βy) = αTx+βTy
T is called linear operator.
<3-3 Bounded linear operator>
Norm space V
Subset of VD
sup{||x|| ; x∈D} < ∞
D is called bounded.
Linear operator from norm space V to norm space V1 T
αi ⊂h*
∈ Z
ij Ki - Ki-1 / qi – qi-1 ( i , j∈I )
p [1-aijp]qiEi1-aij-pEjEip = 0 ( i , j∈I , i ≠j )
