On a Certain Representation in the Pairs of Normed Spaces

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1 Applied Mathematical Sciences, Vol. 12, 2018, no. 3, HIKARI Ltd, On a Certain Representation in the Pairs of ormed Spaces Ahiro Hoshida Suzuya, Chuo-ku, Saitama-shi, Saitama, , Japan Copyright c 2018 Ahiro Hoshida. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract ormed spaces X, Y, a vector space Z, and a bilinear map F : X Y Z induce the projective semi-normed space X F Y := F (ϕ j, ψ j ) n, ϕ 1,, ϕ n X, ψ 1,, ψ n Y. We show that for any element f of X F Y, there exists a representation F (ϕj, ψ j ) satisfying n ϕ 1 X ψ 1 Y n ϕ n X ψ n Y f X F Y. Mathematics Subject Classification: 46A32, 46B28, 44A35, 42A85 Keywords: representation, projective norm, convolution, tensor product 1 Introduction and Preliminaries Let K := R or C be the base field through this paper. Let X and Y be normed spaces, Z be a vector space, and F : X Y Z be a bilinear map. This situation often occurs. For example,

2 116 Ahiro Hoshida Example 1.1 (tensor product). Let X Y the space defined by { } X Y := c j (ϕ j, ψ j ) n, c j K, (ϕ j, ψ j ) X Y. We denote by the relation on X Y, defined by that (ϕ 1 + ϕ 2, ψ 1 ) (ϕ 1, ψ 1 ) + (ϕ 2, ψ 1 ), (ϕ 1, ψ 1 + ψ 2 ) (ϕ 1, ψ 1 ) + (ϕ 1, ψ 2 ), (cϕ 1, ψ 1 ) c(ϕ 1, ψ 1 ) (ϕ 1, cψ 1 ) hold for any ϕ 1, ϕ 2 X, ψ 1, ψ 2 Y, and c K. The quotient space X K Y := (X Y )/ is called a tensor product of X and Y. The quotient map is a bilinear map. F 1 : X Y X K Y Example 1.2 (multiplication). Suppose there is an algebra W satisfying X, Y W. Then the multiplication is a bilinear map. F 2 : X Y W : (ϕ, ψ) ϕ ψ, Example 1.3 (convolution). Let (Ω, µ) be a measure space, 1 p, and = 1. We denote p q Lp be the L p space of functions on (Ω, µ), X := L p, and Y := L q. By the Hölder s inequality, the convolution F 3 : X Y L : (ϕ, ψ) ϕ(x y) ψ(y)dµ, is well-defined and a bilinear. ow, we denote by X F Y the space defined by { } X F Y := F (ϕ j, ψ j ) n, ϕ 1,, ϕ n X, ψ 1,, ψ n Y. For f = F (ϕj, ψ j ) X F Y, we define a semi-norm f X F Y (which is sometimes called the projective norm) on X F Y as { } f X F Y := inf ϕ j X ψ j Y representations of f in X F Y. It is an interesting problem what representations for an element of X F Y we can take. In this paper, we show that for any element f of X F Y, there exists a representation F (ϕj, ψ j ) which satisfies (Theorem 2.2). y Ω n ϕ 1 X ψ 1 Y n ϕ n X ψ n Y f X F Y

3 On a certain representation in the pairs of normed spaces Main Result First, we need the following lemma. Lemma 2.1. For ϕ X and ψ Y, there are Φ X and Ψ Y with F (ϕ, ψ) = F (Φ, Ψ) and Φ X = Ψ Y. Proof. When ϕ 0 and ψ 0, if we set Φ := ψ Y ϕ X ϕ and Ψ := then F (ϕ, ψ) = F (Φ, Ψ) and Φ X = ϕ X ψ Y = Ψ Y. When ϕ = 0 or ψ = 0, let Φ := 0 and Ψ := 0. ϕ X ψ Y ψ, ow, we show the main result. Theorem 2.2. Let ɛ, ɛ > 0. For f X F Y, there is a representation of f, f = 1j F (Φj, Ψ j ) such that f X F Y ɛ < Φ j 2 X = Ψj 2 Y < f X F Y +ɛ for all 1 j. Proof. If we take an f X F Y, then by Lemma 2.1, there is a representation of f, f = F (ϕj, ψ j ) such that f X F Y ϕ j 2 X < f X F Y + ɛ and ϕ j X = ψ j Y for 1 j n. Assume n 2 and ϕ 1 2 X ϕn 2 X. We show that it may assume ϕ 1 2 X > 0 without loss of generality. Indeed, assume ϕ n 2 X > 0 and ϕ1 2 X = 0. We take 2 k n with ϕk 2 X > 0 and ϕ k 1 { 2 X = 0. If we set { ϕ k Φ i := k, 1 i k ψ k ϕ i, k + 1 i n and Ψi := k, 1 i k ψ i, k + 1 i n, then f = F (ϕj, ψ j ) = F (Φj, Ψ j ) and 0 < Φ 1 2 X Φ n 2 X. When f X F Y > ɛ, if we take an ɛ with 0 < ɛ f X F Y f X F 1, then Y ɛ we can take r j Q satisfying { ϕ ϕ j 2 j 2 X X r j < min ( f } X F Y + ɛ) ϕ 1 2 X + +, ϕ j 2 ϕn X(1 + ɛ ) 2 X for 1 j n, and take k j satisfying the ratio for 1 j n. r 1 : : r n = k 1 : : k n

4 118 Ahiro Hoshida Let k 0 := 0 and := 0ln k l. If we set Φ j := ϕi and Ψ j := ψi 1 i n and 0li 1 k l + 1 j 0li 1 k l + k i, then we see that for f = F (ϕ i, ψ i ) = F (Φ j, Ψ j ) 1in 1j and f X F Y ɛ < Φ j 2 X = Ψj 2 Y < f X F Y +ɛ for 1 j. Indeed, for 1 i n and 0li 1 k l + 1 j 0li 1 k l + k i, Φ j 2 X = ϕi 2 X and ri k i k k n + + r n r1, f X F Y ɛ f X F Y (1 + ɛ ) ϕ1 2 X + + ϕn 2 X (1 + ɛ ) (1 + ɛ ) < ϕi 2 X < f X F Y + ɛ hold. When f X F Y ɛ, we can take r j Q satisfying ϕ j 2 X r j < ϕj 2 X ( f X F Y + ɛ) ϕ 1 2 X + + ϕn 2 X for 1 j n, and take k j satisfying the ratio r 1 : : r n = k 1 : : k n for 1 j n. Let k 0 := 0 and := 0ln k l. If we set Φ j := ϕi and Ψ j := ψi, then

5 On a certain representation in the pairs of normed spaces 119 we see that f X F Y ɛ 0 < Φ j 2 X < f X F Y + ɛ hold for 1 i n and 0li 1 k l + 1 j 0li 1 k l + k i. This leads the conclusion. References [1] A. Defant and K. Floret, Tensor orms and Operator Ideals, orth- Holland, Amsterdam, [2] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, Cambridge, Received: January 3, 2018; Published: January 25, 2018