Cmjv04i06p0403

J. Comp. & Math. Sci. Vol.4 (6), 403-406 (2013)

H ö lder Valuation For Vector Spaces M. H. HOSSEINI1 and A. ALLAHYARI2 1

Department of Mathematics, Birjand University, P.O. Box 11111, Birjand, IRAN. 1 Department of Mathematics, Birjand University, P.O. Box 22222, Birjand, IRAN. (Received on: November 21, 2013) ABSTRACT Abstract. The purpose of this article is to introduce the notion (C1, C2)-Hölder valuation on vector space over field and it is proved that a (C1,C2)-Hölder valuation equivalent to classical valuation on vector space over real number. These results provide an extension of the Garsia theorem2 for vector space.

Keywords: Valuation, Hölder valuation, Hölder equivalent, vector space.

1. INTRODUCTION AND PRELIMINARIES

M onto is called a valuation on M, if the following condition are satisfied:

The theory of valuations may be viewed as a branch of topological algebra.

The devolopment of valuation theory has spanned over more than a hundred years.

First introduced the notion of valuations on fields. Details on valuations on fields can be found in many monographs, e.g Endeler1, Ribenboim8, and Schilling9. Then Manis introduced the notion of valuations in the category of commutative rings can be found in Manis6, Huckab5, and Knebusch and Zhang6. Then Zeng Guangxing4 introduced the notion of valuations on a module.

(i) For any x, y ∈ M, ν(x + y) ≥ min{ν(x), ν(y)}; (ii) If ν(x) ≤ ν(y),x, y ∈ M, then ν( x) ≤ ν ( y) for all ∈ R; (iii) Put ν−1(∞) = {x ∈ M|ν(x) = ∞}. If ν( z) ≤ ν( z), where , ∈ R, and z∈ M\ν−1(∞), then ν( x) ≤ ν( y) for all x ∈ M; (iv) For every a ∈ R\(ν−1(∞) : M), there is an ᇱ ∈ R such that ν(( )x) = ν(x).

In this case ∆ is called the value set of ν, and ν−1(∞) is called the core of ν.

Definition 1.1. Let M be an R-module where R is a ring and be and ordered set with maximum element ∞. A mapping ν of

Remark 1.2. In Gravent3 definition of valuation on vector space is adopted as follows. Let V be a nonzero vector space over field F, and let ≤ be and ordering set

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M. H. Hosseini, et al., J. Comp. & Math. Sci. Vol.4 (6), 403-406 (2013)

such that has the minimum element ßł, a mapping ν of V onto is said to be a valuation on V, if the following conditions are satisfied: (V1) For x ∈ V, ν(x) = if and only if x = 0; (V2) For any x, y ∈ V, ν(x + y) ≤ max { ν(x), ν(y) }; (V3) For every nonzero element a F, ν(ax) = ν(x) Let ᇹ be the dual ordering of ≤ such that, for all x,y V, x ᇹ y if y≤ x. Then becomes the maximum elements in with respect to ᇹ . In this case, a mapping ν of V onto is said to be a valuation on V if the following conditions are satisfied: ( 1) For x ∈ V, ν(x) = if and only if x = 0; ( 2) For any x, y ∈ F, ν(x + y) ᇹ min { ν(x), ν(y) }; ( 3) For every nonzero element a F, ν(ax) = ν(x).

In this case such a mapping ν as above is clearly a valuation on V in the sence of the definition 2. For the valuation ν, the core is {0}, and the induced valuation pair is (F, {0} ).

2. HĂśLDER RIGIDITY FOR THE SET OF VECTOR SPACE Definition 2.1. Let V be vector space over field F, and be an ordered set with maximum element ∞, and C1 ≼ 1, C2 ≼ 1. A (C1, C2)-HĂślder valuation on vector space V over field F, is a mapping âˆĽ.âˆĽ : G → satisfying: (H i) For x ∈ V , âˆĽxâˆĽ = ∞ x = 0; (H ii) For any x, y ∈ V, âˆĽx + yâˆĽ ≼ C2min

{ âˆĽxâˆĽ, âˆĽyâˆĽ }; (H iii) For every nonzero element a F and x ∈ V, C1−1âˆĽxâˆĽ ≤ âˆĽaxâˆĽ ≤ C1âˆĽxâˆĽ.

Remark 2.2. We note a (1, 1)-HĂślder valuation on vector space V over field F is a classical valuation on vector space V over field F. Definition 2.3. Let |.|1, |.|2 be two valuation on vector space V over field F. Then we say that they are (C0, Îą)-HĂślder equivalent, if for all x∈ V, C0 ≼ 1, Îą > 0 C0−1|x| ŕ°ˆŕŹľ ≤ |x|2 ≤ C0|x| ŕ°ˆŕŹľ where ι’= Îą or Îą' = Îąâˆ’1.

HĂśLDER VALUATION FOR VECTOR SPACE Theorem 2.4. let | . |1 : V → R+ âˆŞ { ∞ } be a (C1, C2)-HĂślder valuation on vector space V over real number field R, where C1 ≼ 1,C2Îą ≼ 4. Then there exists a classical valuation |.|1 on vector space V over real number field R, that is (2, Îą)-Holder equivalent to the (C1, C2)-Holder valuation |.|2 , where Îą = log2C1.

For the proof of the theorem we need several lemmas and propositions.

Definition 2.5. For x ∈ V, we define |x|3 = |x| ŕ°ˆŕŹľ .

Lemma 2.6. The mapping |.|3: V→ R+âˆŞ{∞} is a (2, C2Îą)-HĂślder valuation on vector space V over real number field R. Proof. (1): For x ∈ V , |x|3 = |x| ŕ°ˆŕŹľ = ∞ |x|1 = ∞ x = 0 (2): If x, y ∈ V, then

Journal of Computer and Mathematical Sciences Vol. 4, Issue 6, 31 December, 2013 Pages (403-459)

M. H. Hosseini, et al., J. Comp. & Math. Sci. Vol.4 (6), 403-406 (2013)

x + y|3 = |x + y| ఈଵ ≥ (C2min{|x|1, |y|1})α.

Therefore |x + y|3 ≥ (C2αmin{|x| ఈଵ , |y| ఈଵ } = C2αmin { |x|3, |y|3 } ).

(3): For any nonzero element R, we have C1−1|x|1 ≤ | x|1 ≤ C1|x|1.

Hence C1−α|x| ఈଵ ≤ (| x|3 = | x| ఈଵ ) ≤ C1α |x| ఈଵ .

Therefore 2−1|x|3 ≤ | x|3 ≤ 2|x|3.

Lemma 2.7. Let x ∈ V and { n } = { |nx|3 }. Then the sequence { n} is bounded.

Proof. By part (3) of lemma 2.6, we have 2−1|x|3 ≤ |nx|3 ≤ 2|x|3.

Therefore the sequence { n} is bounded.

Now we prove that the sequence { n} is increasing. n+1 = |(n + 1)x|3 = |nx + x|3 ≥ C2αmin{ |nx|3, |x|3 } ≥ C2αmin{ |nx|3, 2−1|nx|3}(by part (2),(3) of lemma 2.6).

Therefore n+1 ≥ C2α2−1|nx|3. Since C2α ≥ 4 ≥ 2 and so C2α2−1 ≥ 1.

Consequently n+1 ≥ ( |nx|3 = n ).

Therefore { n} is increasing. Hence we have the following results.

Corollary 2.8. The sequence { n} = { |nx|3 } is convergent sequence.

Proof. The sequence { n} = { |nx|3 } is bounded and increasing on R+ since R+ is metric space. Therefore the sequence { n} = { |nx|3 } is convergent sequence.

Proposition 2.9. Let |x|2 = limn→∞ |nx|3.

Then 2−1|x|3 ≤ |x|2 ≤ 2|x|3. Proof. By part (3) of lemma 2.6, we have

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2−1|x|3 ≤ |nx|3 ≤ 2|x|3.

Therefore lim 2−1|x|3 ≤ limn→∞|nx|3 ≤ lim 2|x|3. n→∞

n→∞

Hence 2−1|x|3 ≤ |x|2 ≤ 2|x|3.

Proposition 2.10. The mapping | . |2 : G → R+ ∪ {∞} ∪ {∞}by equation |x|2 = lim |nx|3 n→∞

is classical valuation on vector space V over real number field R. Proof. (1) By proposition 2.9, for x ∈ V, we have |x|2 = ∞ if and only if |x|3= ∞ if and only if x = 0 (by part (1) of lemma 2.6 ).

(2) If x, y ∈ V, then |x + y|3 ≥ C2αmin{ |x|3, |y|3 } ≥ 2min{ 2|x|3, 2|y|3 }.

Thus by proposition 2.9, we have |x + y|3 ≥ 2min{|x|2, |y|2}.

On the other hand |x + y|2 ≥ 2−1|x + y|3 ≥ 2−1( 2min{ |x|2, |y|2 } ).

Therefore |x + y|2 ≥ min { |x|2, |y|2 }.

3) For any nonzero element R, and x ∈ V, we have | x|2 = lim |n x|3 = lim |n x|3 = |x|2. n →∞

n |a| →∞

Therefore

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M. H. Hosseini, et al., J. Comp. & Math. Sci. Vol.4 (6), 403-406 (2013)

|.|2 is classical valuation on vector space V over real number field R. Proof of Theorem 2.4. By proposition 2.10, the mapping |.|2 : V → R+ ∪{∞} is classical valuation on vector space V over real number field R , and by proposition 2.9, we have 2−1|x|3 ≤ |x|2 ≤ 2|x|3.

Therefore 2−1|x| ఈଵ ≤ |x|2 ≤ 2|x| ఈଵ .

Thus classical valuation |.|2 on vector space V over real number field R that is (2, α)-Hölder equivalent to the (C1, C2)-Hölder valuation | . |1 on vector space V over real number field R. REFERENCES 1. O. Endler, valuation Theory. New York: Springer-Verlag (1972).

2. E. Munoz. Garcia, Holder absolute values are equiavalent to classical, ones Proc. Amer. Math. Soc. 127, No. 7,

1967-1971 (1999).

3. K. A. H. Gravett, Valued linear spaces.

Quart. J. Math. Oxford 6: 309-315 (1955).

4. Zeng. Guangxing, valuations on a module, Comm. Algebra, 35 (8), 23412356 (2007).

5. J.A. Huckaba, Commutative Rings with zero divisors, New York: Marcel Dekker, Inc (1988).

6. M. Knebusch, D. Zhang, Manis valuations and Prufer Extensions I. A new chapter in Commutative algebra.

Lecture Notes in Math. 1791. Berlin: Springer-Verlage (2002).

7. M. Manis, valuations on a commutative ring. Proc. Amer. Marh Soc. 20:193-198 (1969).

8. P. Ribenboim. The Theorie des valuations. Montreal: Less presses de I’Universite de Montreal (1964).

9. O.F.G. Schilling, The Theory of valuations New York: Amer. Math. SOC Thomas, J. (1978). Set Theory. New York: Academic press, Inc (1952).

Journal of Computer and Mathematical Sciences Vol. 4, Issue 6, 31 December, 2013 Pages (403-459)