Ann. Funct. Anal. 4 (2013), no. 1, A nnals of F unctional A nalysis ISSN: (electronic) URL:

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1 Ann. Funct. Anal. 4 (013), no. 1, A nnals of F unctional A nalysis ISSN: (electronic) URL: A CHARACTERIZATION OF THE INNER PRODUCT SPACES INVOLVING TRIGONOMETRY DAN ŞTEFAN MARINESCU 1, MIHAI MONEA, MIHAI OPINCARIU 3 AND MARIAN STROE 4 Communicated by J. Chmieliński Abstract. In this paper we will give a new characterization of the inner product space which use the trigonometry. We conclude that a normed space (X, ) is an inner product space if and only if there exists α R\πQ so that for any x, y X. x cos α + y sin α + y cos α x sin α = x + y, 1. Introduction The problem of finding some necessary and sufficient geometric conditions for a normed space to be an inner product space has been investigated by several mathematicians for a long time. Some characterizations of inner product spaces and their generalizations can be found in [1,, 4] and references therein. In [3], Moslehian and Rassias gave a characterization of the inner product space using an Euler-Lagrange type identity. The result is presented in the next proposition. Proposition 1.1. Let be (X, ) a normed space. Then the norm is derived from an inner product space if and only if ax + by + bx ay = ( a + b ) ( x + y ), for any x, y X and a, b > 0. Date: Received: 31 August 01; Accepted: 0 October 01. Corresponding author. 000 Mathematics Subject Classification. Primary 46C15; Secondary 46B0. Key words and phrases. Normed spaces, inner product spaces, trigonometry. 109

2 110 D. Ş. MARINESCU, M. MONEA, M. OPINCARIU, M. STROE Starting from this result, we will obtain another characterization for the inner product space involving the trigonometry.. Some Preliminary Results The identity from previous proposition could be transformed. After we divide with a + b, we obtain a a + b x + b a + b y + b a + b x a a + b y = x + y. But, it easy to see that it exists a real number α ( ) 0, π for which cos α = a a and sin α = b +b a. In this context, our identity becomes +b x cos α + y sin α + x sin α y cos α = x + y. In fact, the result is more general and it is exposed and proved in the next proposition. Proposition.1. Let be (X, ) a normed space. Then the norm is derived from an inner product space if and only if x cos α + y sin α + y cos α x sin α = x + y, for any x, y X and α R. Proof. If X is an inner product spaces, we have x cos α + y sin α + y cos α x sin α = x cos α + y sin α, x cos α + y sin α + y cos α x sin α, y cos α x sin α = x cos α + x cos α, y sin α + y sin α, x cos α + y sin α + + y cos α x sin α, y cos α y cos α, x sin α + x sin α = ( sin α + cos α ) x + ( sin α + cos α ) y + ( x, y + y, x x, y y, x ) sin α cos α = x + y. For the second part of the proof, we choose α = π 4 and identity x cos α + y sin α + x sin α y cos α = x + y becomes x + y which conclude our proof. + x y = x + y,

3 A CHARACTERIZATION OF THE INNER PRODUCT SPACES 111 But, our scope is to improve this result. For this we will remind two classical result from the mathematical analysis. First, we denote C (O, 1) the unit circle from R, and πq = {πk k Q}. Lemma.. For any α R\πQ, the set {(cos nα, sin nα) n N} is dense in C (O, 1). Lemma.3. Let be (X, ) a normed space. The the function f : X R, f (x) = x is continuous. Now we can present and prove the main result of our paper. 3. The Main Result Using the lemmas reminded in previous paragraph, now we can improve the results from Proposition.1. in the next form: Theorem 3.1. Let be (X, ) a normed space. Then the norm is derived from an inner product space if and only if it exists α R\πQ so that for any x, y X. x cos α + y sin α + y cos α x sin α = x + y, Proof. The only if part of the proof is true for any α R how we seen in Proposition.1., so particullary for an α R\πQ. For the if part, we consider that there exists α R\πQ with x cos α + y sin α + y cos α x sin α = x + y, for any x, y X. We replace x with x cos α + y sin α and y with y cos α x sin α. We obtain x cos α + y sin α + x sin α y cos α = = (x cos α + y sin α) cos α + (y cos α x sin α) sin α + + (y cos α x sin α) cos α (x cos α + y sin α) sin α = x ( cos α sin α ) + y sin α cos α + y ( cos α sin α ) x sin α cos α So, we obtain the identity = x cos α + y sin α + y cos α x sin α. x cos α + y sin α + y cos α x sin α = x + y, for all x, y X. Further, we will use the mathematical induction and we suppose that the identity x cos kα + y sin kα + y cos kα x sin kα = x + y,

4 11 D. Ş. MARINESCU, M. MONEA, M. OPINCARIU, M. STROE is true for some k N and we prove it for k + 1. In the initial identity, we replace x with x cos kα + y sin kα and y with y cos kα x sin kα and we have x cos kα + y sin kα + y cos kα x sin kα = = (x cos kα + y sin kα) cos α + (y cos kα x sin kα) sin α + + (y cos kα x sin kα) cos α (x cos kα + y sin kα) sin α = x (cos kα cos α sin kα sin α) + y (sin kα cos α + cos kα sin α) + + y (cos kα cos α sin kα sin α) x (sin kα cos α + cos kα sin α) = x cos (k + 1) α + y sin (k + 1) α + y cos (k + 1) α x sin (k + 1) α. So, we have that the identity x cos nα + y sin nα + y cos nα x sin nα = x + y, is true for all n N. Now, we apply Lemma.. There exists a sequence (a n ) n N of natural numbers for which and Then, we obtain lim a n = n lim (cos a nα, sin a n α) = n (, ). x cos a n α + y sin a n α + y cos a n α x sin a n α = x + y. If we make n and use Lemma.3., we have x + y + x y for all x, y X and our proof is ready. = x + y, Final remark: Obviously, the set R\πQ can be enlarged. For example, if α { ( k+1 π : k Z, n N}, then (sin nα, cos nα) = ±, ± ) and from the 4n identity x cos nα + y sin nα + y cos nα x sin nα = x + y, we get the parallelogram identity. The question is: what is the biggest set A, such that and for which the theorem holds true. R\πQ A R

5 A CHARACTERIZATION OF THE INNER PRODUCT SPACES 113 References 1. C. Alsina, J. Sikorska and M.S. Tomás, Norm derivatives and characterizations of inner product spaces, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, D. Amir, Characterizations of inner product spaces, Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, M.S. Moslehian and J.M. Rassias, A characterization of inner product spaces concerning an Euler-Lagrange identity, Commun. Math. Anal. 8 (010), no., K. Nikodem and Z. Páles, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal. 5 (011), no. 1, National College IANCU DE HUNEDOARA, Hunedoara, Romania address: marinescuds@gmail.com National College DECEBAL, Deva, Romania address: mihaimonea@yahoo.com 3 National College AVRAM IANCU, Brad, Romania address: opincariumihai@yahoo.com 4 Economic College EMANOIL GOJDU, Hunedoara, Romania address: maricu stroe@yahoo.com