Normed linear spaces

The set of bounded operators acting in a Hilbert space form a normed linear space. The norm is given by the bound on the operator... 

This set also forms a normed linear space with norm defined by... 

UNBOUNDED LINEAR OPERATORS Theory and Applications, Seymour Goldberg. Classic presents systematic treatment of the theory of unbounded linear operators in normed linear spaces with applications to differential equations. Bibliography. 199pp. 5)4 x 8)4. 64830-3 Pa. 7.00... 

Thus we can see that the linear vector space contains another very important property of the Euclidean space it has a basis. However, there is no distance in a linear vector space. It would be extremely useful if we could combine these two properties of the Euclidean space, a distance and a basis, within one space. This space is called a normed linear space. 

A normed linear space is a linear space N in which for any vector f there corresponds a real number, denoted by f and called the norm of f, in such a manner that... 

Let X and Y be normed spaces with the same system of scalars. There is a very important class of operators in the normed linear spaces which are called linear operators. 

Assume that X and Y are two Banach spaces (complete normed linear spaces) and A is some operator from X to T. 

In this section we will prove that the Lieb functional is differentiable on the set of E-V-densities and nowhere else. The functional derivative at a given E-V-density is equal to — v where v is the external potential that generates the E-V-density at which we take the derivative. To prove existence of the derivative we use the geometric idea that if a derivative of a functional G[n] in a point n0 exists, then there is a unique tangent line that touches the graph of G in a point (n0, G[ 0 ). To discuss this in more detail we have to define what we mean with a tangent. The discussion is simplified by the fact that we are dealing with convex functionals. If G B — TZ is a differentiable and convex functional from a normed linear space B to the real numbers then from the convexity property it follows that for n0,nj 5 and 0 < A < 1 that... 

Linear operators. Let X and Y be normed vector spaces and T be a subspace of the space X. If to each vector x V there corresponds by an... 

It is worth noting here that in a finite-dimensional space any linear operator is bounded. All of the linear bounded operators from X into Y constitute what is called a normed vector space, since the norm j4 of an operator A satisfies all of the axioms of the norm ... 

Stability of a difference scheme. Let two normed vector spaces and be given with parameter h being a vector of some normed space with the norm /i > 0. In dealing with a linear operator with the domain V Ah) — and range TZ Af ) C B we consider the equation... 

According to von Neumann (9), an abstract Hilbert space X is a linear space A = x having a binary product , which satisfies the conditions, Eqs. (1.8)—(1.10), and which further contains all its limit points in the norm x = 1/2 and is separable. The last assumption means that there exists an enumerable set Jf" = xk which is everywhere dense in and which ensures the existence of at least one complete orthonormal set

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Thus we can see that a normed linear vector space contains both a basis and a distance. It has two important properties of the Euclidean space, but not all of its properties. One property, which is still missing, is the analogue of the dot product of two vectors in the Euclidean space. This property is very important, because it actually provides the possibility not only to determine the distance between two points, but also to characterize qualitatively the direction from one point to another in abstract mathematical space. Therefore, the geometrical properties of the space become more rigid. We introduce the space with these properties below. 

It is easy to verify that the norm introduced in (A.59) satisfies all required conditions for the norm of the normed space, and I leave this proof to the reader as an exercise. This norm is called the uniform norm. The linear space of continuous functions, Co [a, 6], equipped with the uniform norm, forms a normed space, denoted by C [a, b]. The distance between two functions, /(x) and g x), in the space C[a,b] with the uniform norm is equal to... 

Thus, now two functions will be close to each other if the integral of thcii difference is small enough. The presence of one or two outliers will not affect the result significantly (see Figure A-5). The linear space of continuous functions on the real interval [a, 5] equipped with the Lj norm, is called the Li[a, 6] space. This is a linear normed space, but it is not a Hilbert space because it has no inner product operation. 

The linear space of functions, integrable on the real interval [a, 6] and equipped with the L2 norm, is called L2[a, 6] space. A linear normed space L2[a, b] is a Hilbert space, and therefore, possesses all the properties of the Hilbert space discussed above. 

Let us start by defining what we mean with a functional derivative. The derivative we will talk about is what in the mathematical literature [9-14] is refered to as a Gateaux derivative. Let G B —<> TZ be a functional from a normed function space B to the real numbers TZ. If for every h G B there exist a continuous linear functional 8G/8/ B—>11 defined by ... 

Now the functional on the right hand side of the inequality sign is, for a given v, a linear functional of n. The inequality sign tells us that this functional lies below the graph of El[w]. A linear functional with this property is called FL-bounded. Let us give a general definition of these linear functionals. Let F be a functional F B — 1Z from a normed function space (a Banach space) B to the real numbers. Let B be the dual space of B, i.e., the set of continuous linear functionals on B. Then L E B is said to be /"-bounded if there is a constant C such that for all n G B... 

Let V be a normed space, and J V —> i be an arbitrary functional. We assume that there exists a linear and continuous functional such that for each u G V... 

Linear bounded operators in a real Hilbert space. Let H he a real Hilbert space equipped with an inner product x,y) and associated norm II X II = (x, x). We consider bounded linear operators defined on the space... 

Let A be a positive self-adjoint linear operator. By introducing on the space H the inner product x,y) = Ax,y) and the associated norm x) we obtain a Hilbert space Ha, which is usually called the energetic space Ha- It is easy to show that the inner product... 

Linear operators in finite-dimensional spaces. It is supposed that an n-dimensional vector space R is equipped with an inner product (, ) and associated norm a = / x x). By the definition of finite-dimensional space, any vector x G i n can uniquely be represented as a linear combina-... 

Operator-difference schemes. We now consider a linear system 8h depending on a parameter h as a vector of some normed space equipped with the norm h. With regard to the linear system 8h, it is reasonable to introduce a collection of norms li 111,. II II/, m II ll m. .., thus causing... 

Sufficient stability conditions for two-layer schemes in linear normed spaces. We now raise the question concerning sufficient stability conditions for two-layer schemes in linear normed spaces. In full details these investigations will be carried out in Section 2 for the case when Bh = Hh is a real Hilbert space. 

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