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1 Introduction A FORMULAS FOR THE OPERATOR NORM AND THE EXTENSION OF A LINEAR FUNCTIONAL ON A LINEAR SUBSPACE OF A HILBERT SPACE 1; Halil Ibrahim· Çelik 1 Marmara University, Faculty of Arts and Sciences, Department of Mathematics, Istanbul,· TURKEY E-mail: [email protected] ( Received: 25.11.2019, Accepted: 18.12.2019, Published Online: 31.12.2019) Abstract In this paper, formulas are given both for the operator norm and for the extension of a linear functional de…ned on a linear subspace of a Hilbert space and the results are illustrated with examples. Keywords: Linear functional; Cauchy-Schwarz inequality; norm; operator norm; Hilbert spaces; Hahn-Banach theorem; Riesz representation theorem; orthogonal vectors; orthogonal decomposition. MSC 2000: 31A05; 30C85; 31C10. 1 Introduction Linear functionals occupy quite important place in mathematics in terms of both theory and applica- tion. The weak and weak-star topologies, which are fundamental and substantial subject in functional analysis, are generated by families of linear functionals. They are important in the theory of di¤eren- tial equations, potential theory, convexity and control theory [6]. Linear functionals play fundamental role in characterizing the topological closure of sets and therefore they are important for approxima- tion theory. They play a very important role in de…ning vector valued analytic functions, generalizing Cauchy integral theorem and Liouville theorem. Therefore the need arises naturally to construct lin- ear functionals with certain properties. The construction is usually achieved by de…ning the linear functional on a subspace of a normed linear space where it is easy to verify the desired properties and then extending it to the whole space with retaining the properties. This is not always easy in the case of general normed linear spaces. We specialize to linear functionals de…ned on the subspaces of a Hilbert space and provide formulas (Theorem 2) both for the operator norms and norm preserving linear extensions of linear functionals. We start with basic de…nitions and results and …xed notations that will be used in the sequel. We denote the …eld of the real numbers R or the …eld of the complex numbers C by F. We denote the absolute value function by : de…ned on the …eld F. So for x R, if x < 0 then x = x and if x 0 j j 2 j j then x = x. For z = x + iy C we have z = x2 + y2. The complex number z = x iy is the complexj j conjugate of the number2 z = x + iyj. j p De…nition 1.1. Let X be a linear space over the …eld F and : : X R be a function. If the function : satis…es the following properties k k 7! k k 1. 0 = 0 and x > 0 for every x X 0 (positivity de…niteness); k k k k 2 n f g Ikonion Journal of Mathematics 2019, 1(2) 2. x = x for every F and x X (homogeneity); and k k j j k k 2 2 3. x + y x + y for all x; y X (triangular inequality); k k k k k k 2 then the function : is called a norm on the space X and the pair (X; : ) is called a normed linear space. k k k k n n Example 1.2. Let p [1; ) and R = x = (x1; : : : ; xn): xj R for j = 1; : : : ; n . R is 2 1 f 2 g a linear (vector) space over the …eld R with componentwise addition and scaler multiplication. For 1=p n n p x = (x1; : : : ; xn) R de…ne x p = j=1 xj and x = max xj : j = 1; : : : ; n . Then for 2 k k j j k k1 fj j g 1 p the functions : are norms on the space n. Hence ( n; : ) is a normed linear spaces p P R R p for each 1p [1; ]. k k k k 2 1 Inner products spaces are very important sources of normed linear spaces. De…nition 1.3. Let X be a linear space over the …eld F. If the function (:; :): X X F satis…es the following properties 7! 1. (x; x) 0 for all x X and (x; x) = 0 if and only if x = 0; 2 2. (x; y) = (y; x) for every x; y X; 2 3. (x + y; z) = (x; z) + (y; z) for every x; y; z X; and 2 4. ( x; y) = (x; y) for every F and for every x; y X; 2 2 then the function (:; :) is called an inner product on X and the pair (X; (:; :)) is called a inner product space over the …eld F. The number x = (x; x) is called the norm of the vector x X. If x; y X and (x; y) = 0 then the vectors x andk ky are called orthogonal vectors. 2 2 p The inner product generates the most important inequality in mathematics, namely the Cauchy- Schwarz inequality. Theorem 1.4. [Cauchy-Schwarz inequality][2, 4] Let (X; (:; :)) be an inner product space over the …eld F. Then for every x; y X, (x; y) (x; x) (y; y) = x y . The equality occurs if and only if the vectors x and y are linearly2 j dependent.j k k k k p p From the Cauchy-Schwarz inequality it follows that the function x = (x; x) is a norm on the space X. This norm is called the norm generated by the inner productk k function (:; :). If the normed p linear space (X; : )) is a Banach space, that is, if every Cauchy sequence in X converges to a point in X, the inner productk k space (X; (:; :)) is called a Hilbert space. n n n Example 1.5. For x = (x1; : : : ; xn); y = (y1; : : : ; yn) R the function (:; :): R R R n 2 n 7! de…ned by (x; y) = j=1 xjyj is an inner product on the space R . The norm generated by this inner 1=2 n product is the EuclideanP norm x = (x; x) = x2 . The inner product space (Rn; : ) is k k2 j=1 j k k a Hilbert space. p P On normed linear spaces the primary objects of study are the linear operators and linear functionals which play central role in functional analysis. De…nition 1.6. Let (X; : ) and (Y; : 0) be normed linear spaces over the same …eld F and k k k k T : X Y be a mapping. If T ( x + y) = T (x) + T (Y ) for all ; F and x; y X then T is called a7! linear operator. A linear operator T is called bounded if there is2 a real constant2 M > 0 such that T (x) 0 M x for all x X. If T is a bounded linear operator the number T = T = k k k k 2 k kop k k 47 Ikonion Journal of Mathematics 2019, 1(2) inf M : T (x) 0 M x for all x X is called a operator norm of T . The equivalent de…nition of operatork normk is givenk byk the formulas2 T (x) 0 T = T = sup k k : x = 0 = sup T (x) 0 : x 1 k kop k k x k k 6 k k k k k k = sup T (x) 0 : x = 1 k k k k The following is a very useful result for the computation of operator norms of linear operators. Lemma 1.7. [Computation of operator norm] Let (X; : ) and (Y; : 0) be normed linear spaces k k k k over the same …eld F, T : X Y be a bounded linear operator and M 0 be a real constant. If for 7! every x X, T (x) 0 M x and T (x0) 0 = M x0 for a vector x0 X 0 then the operator norm of 2T is kT =k M . k k k k k k 2 n f g k kop Proof. If for every x X, T (x) 0 M x then by the de…nition of operator norm T M. 2 k k k k k kop On the other hand if for a vector x0 X 0 , T (x0) 0 = M x0 then by the de…nition of operator 2 n f g k k k k norm we have M x0 = T (x0) 0 T x0 so that M T . Therefore T = M. k k k k k kop k k k kop k kop Remark 1.8. We note that in the …nite dimensional case the operator norm can be computed by the method of Lagrange multipliers with constraints. It is now a classical result that a linear operator is bounded if and only if it is continuous. The set (X; Y ) of bounded linear operators is a linear space over F with pointwise addition and scaler multi- B plication and : op is a norm on (X; Y ). If (Y; : 0) is a Banach space then the space ( (X; Y ); : op) is a Banach space.k k B k k B k k A bounded linear operator f :(X; : ) (F; : ) is called a bounded linear functional. The k k 7! j j Banach space (X; : op) of bounded linear functional is called dual or conjugate space of the normed linear space (X; : k):k k k Example 1.9. Let (X; (:; :)) be an inner product space over the …eld F and a X be a …xed 2 vector. Then f :(X; : ) (F; : ); f(x) = (x; a) is a bounded linear functional and f = a : k k 7! j j k kop k k Linear functionals are important in terms of generating and characterizing linear subspaces.
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