Journal of Applied Sciences

Year: 2011 | Volume: 11 | Issue: 16 | Page No.: 3039-3043
DOI: 10.3923/jas.2011.3039.3043

Best Approximation in Quotient Generalized 2-normed Spaces

M. Abrishami Moghaddam and T. Sistani

Abstract: The aim of this article is to generalize the best approximation problem to the case of quotient generalized 2-normed spaces. At first, a quotient 2-norm is introduced and with an example we show that the quotient 2-norm is a generalized 2-norm that it is not a 2-norm. Afterward some theorems of approximation in quotient spaces are extended to this newly introduced quotient generalized 2-normed space in order to assess their validity. 2000 Mathematics Subject Classification. Primary 46A12.

Keywords: Generalized 2-normed space, 2-proximinal, 2-semi chebyshev, 2-chebyshev and 2-best approximation

INTRODUCTION

Approximation theory has many important applications in various areas of functional analysis, computer science, numerical solutions of differential and integral equations. As a generalization of normed spaces is 2-normed spaces that play a very important role in functional analysis. What we offer in this study is to investigate approximation theory in generalized 2-normed spaces. The concept of linear 2-normed spaces was introduced by Gahler (1964) as an interesting non-linear generalization of a normed linear space which was developed extensively in different subjects by others. During 1999-2006 (Lewandowska, 1999, 2001, 2003a, 2003b, 2004; Lewandowska et al., 2006) has published a series of papers on 2-normed sets and generalized 2-normed spaces. Lal and Das (1982), Chen (2002) and Rezapour (2009) carry on the development of this concept to 2-functional and approximation in 2-normed spaces.

Let X be a linear space of dimension greater than 1 over K, where K is the field of real or complex numbers. Suppose ||.,.|| be a nonnegative real-valued function on (XxX) satisfying the following conditions:

Then ||.,.|| is called a 2-norm on X and (X, ||.,.||) is called a linear 2-normed space. Every 2-normed space is a locally convex topological vector space. In fact for a fixed bεX, p_b (x) = ||x, b|| is a seminorm and the family P = p_b: bεX of seminorms generates a locally convex topology on X.

Definition 1: Let X and Y be linear spaces, D be a nonempty subset of XxY such that for every xεX and yεY, the sets:

are linear subspaces of the spaces Y and X, respectively. A function ||.,.||: D→[0, ∞) is called a generalized 2-norm on D if it satisfies the following conditions:

Then (D, ||.,.||) is called a 2-normed set. In particular, if D = XxY, (XxY, ||.,.||) is called a generalized 2-normed space. Moreover, if X = Y, then generalized 2-normed space is denoted by (X, ||.,.||).

Definition 2: Let X be a real linear space. Denote by X a non empty subset XxX with the property X = X^-1 and such that the set X^y = {xεX; (x; y)εX} is a linear subspace of X, for all yεX. A function ||.,.||: X→[0, ∞) satisfying the following conditions:

will be called a generalized symmetric 2-norm on X. The set X is called a symmetric 2-normed set. In particular, if X =XxX, the function ||.,.|| will be called a generalized symmetric 2-norm on and the pair (X; ||.,.||) a generalized symmetric 2-normed space.

Every 2-normed space is a generalized 2-normed space but the converse is not true. The following example is a generalized 2-normed space that is not a 2-normed space.

Example 1: Let X be a real linear space having two norms ||.||₁ and ||.||₂. Then (X, ||.,.||) is a generalized 2-normed space with the 2-norm defined by:

Specially if ||.||₁ = ||.||₂, our generalized 2-normed space will be a generalized symmetric 2-normed space.

For further examples see (Lewandowska, 1999, 2001, 2003a, 2003b, 2004; Lewandowska et al., 2006).

BEST APPROXIMATION IN QUOTIENT GENERALIZED 2-NORMED SPACE

In the following theorem a generalized 2-norm on the space:

is defined.

Theorem 1: Let (XxY, ||.,.||) be a generalized 2-normed linear space and G₁ and G₂ be subspace of X and Y, respectively. Define:

for every xεX and yεY. Then |||.,.||| is a generalized 2-norm on.

Proof: First we prove that |||.,.||| is well defined.

If x₁ +G₁ = x₂ + G₂ and y₁ + G₂ = y₂+G₂, then there exists (g, g’)εG₁xG₂ such that x₁ = x₂ + g and y₁ = y₂ + g’. So:

Step 1:

Step 2:

Step 3:

Step 4: Similar to Step 3 we can show that:

Example 2: Let X = Y = R² , G₁ = x axis and G₂ = y axis and let:

where, x = (x₁, x₂), y = (y₁, y₂). It is a generalized 2-norm on R². Then:

It is obvious that |||.,.||| is a generalized 2-norm that it is not a 2-norm.

Definition 3: Let (XxY, ||.,.||) be a generalized 2-normed space, G₁ be a subspace of X and let G₂ be a subspace of Y . Then, G₁xG₂ is called 2-proximinal if for every (x, y)ε XxY there exists (g₀, g₀’) such that:

In this case, (g₀, g₀’) is called 2-best approximation of (x, y) in G₁xG₂ and the set of all 2-best approximations of (x, y) in G₁xG₂ is denoted by . G₁xG₂ is called 2-semi chebyshev if is at most singleton for each (x, y)εXxY and 2-chebyshev if is exactly singleton for each (x, y)εXxY.

Theorem 2: Let (XxY, ||.,.||) be a generalized 2-normed linear space, K₁, G₁ and K₂, G₂ be subspaces of X and Y, respectively such that K₁⊂G₁ and K₂⊂G₂. Then the following are true:

Proof
Step 1: If (g₁ + K₁, g₂ + K₂) is not a 2-best approximation to (x + K₁, y + K₂) in , then:

Hence, there exists (g₀, g₀’)εG₁xG₂ such that |||x-g₀+K₁, y-g₀’ + K₂|||<|||x-g₁ + K₁, y-g₂+K₂|||. Since |||x-g₁+K₁, y-g₂+K₂|||≤||x-g₁, y-g₂, we have |||x-g₀+K₁, y-g₀’+K₂|||<||x-g₁, y-g₂||. Thus, for some (k₁, k₂)εK₁xK₂, |||x-g₀+k₁, y-g₀’+k₂|||<||x-g₁, y-g₂||. K₁⊂G₁ and K₂⊂G₂ implies that (g₀+k₁, g₀’+k₂)εG₁xG₂.Therefore, (g₁, g₂) is not a 2-best approximation to (x, y) in G₁xG₂ and this is a contradiction.

Step 2: By hypothesis:

and |||x-g₁+K₁, y-g₂+K₂|||≤|||x-g’₁+K₁, y-g’₂+K₂||| for all (g’₁, g’₂)εG₁xG₂. Therefore for any (g’₁, g’₂)εG₁xG₂:

Therefore:

Step 3: It is clear by step 2.

Step 4: Let the conclusion be false, then there exists (x, y)εXxY that has two distinct-best approximation (g₁, g₂) and (g₃, g₄) from (G₁xG₂). By Step 1, (g₁+K₁, g₂+K₂) and (g₃+K₁, g₄+K₂) are 2-best approximations to (x+G₁, y+G₂) from . Since is 2-semi-chebyshev in , we have (g₁+K₁, g₂+K₂) = (g₃+K₁, g₄+K₂). Therefore, there exists k₁εK₁ and k₂εK₂ such that g₃ = g₁ + k₁, g₄ = g₂+K₂ and (k₁, k₂)≠(0, 0). Thus:

Therefore (k₁, k₂) and (0, 0) are 2-best approximations to (x-g₁, y-g₂) from (K₁xK₂). Since (k₁, k₂)≠(0, 0), then (K₁xK₂) is not 2-semi-chebyshev in XxY and this is a contradiction.

Step 5: It is an immediate conclusion from step 1.

Step 6: It is clear by step 3 and 1.

Step 7: If it is not, then there exists , (g₁+K₁, g₂+K₂) and (g₃+K₁, g₄+K₂) belong to such that (g₁+K₁, g₂+K₂)≠(g₃+K₁, g₄+K₂). Thus, g₁-g₂∉K₁ or g₃-g₄∉K₂. Since (K₁xK₂) is 2-proximinal in XxY, then and . Let and by step 2, (g₁+k₁, g₂+k₂) and (g₃+k’₁, g₄+k’₂) are 2-best approximation to (x, y) from G₁xG₂. Since G₁xG₂ is 2-semi-chebyshev, then g₁+k₁ = g₃+k’₁ and g₂+k₂ = g₄+k’₂. Hence, (g₁-g₃+g₂-g₄) and this is a contradiction.

Step 7: It is clear by step 6 and 7.

Let G₁xG₂ be a 2-proximinal subspace of a generalized 2-normed linear space XxY and let denote the quotient space equipped with the generalized 2-norm |||.,.||| defined above. Then we define the quotient map:

by π (x, y) = (x+G₁, y+G₂).

Theorem 3: Let G₁xG₂ and K₁xK₂ be subspaces of a generalized 2-normed linear space XxY and K₁xK₂. If K₁xK₂⊂G₁xG₂ be 2-proximinal in G₁xG₂, then:

Furthermore if K₁xK₂ be 2-proximinal in XxY, then:

Proof: By part (1) of theorem (2), it is clear that:

Now let K₁xK₂ be 2-proximinal in XxY. By part (2) of theorem (2), if:

and:

then:

Therefore:

Hence,

Theorem 4: For a linear subspace G₁xG₂ of a generalized 2-normed linear space XxY, the following statements are equivalent:

We have:

Proof: (1)⇔(2) If G₁xG₂ is 2-proximinal, (x, y)εXxY (G₁xG₂) and , then:

conversely, if we have (2) and (x, y)εXxY then (x, y) = (g₁+x₁, g₂+y₁), where, (g₁, g₂)εG₁xG₂ and . Hence:

Hence:

(1)⇔(3) If G₁xG₂ be 2-proximinal and:

Then:

and π (x-g₁, y-g₂) = (x+G₁, y+G₂). Conversely, if we have (3) and (x, y)εXxY, then:

So (x+G₁, y+G₂) = π (x₁, y₁), where . Hence, (x-x₁, y-y₁) = (g₁, g₂)ε,₁xG₂ and:

So . Which this complete the proof.

CONCLUSION

In this study we introduce the concept of best approximation in generalized quotient 2-normed spaces and present some results.

REFERENCES