scholarly journals A generalization of Lagrange multipliers

1970 ◽  
Vol 3 (3) ◽  
pp. 353-362 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Banach Spaces ◽  
Lagrange Multipliers ◽  
Lagrange Equation ◽  
Linear Mapping ◽  
Real Field ◽  
Dimensional Problem ◽  
Continuous Linear ◽  
Continuous Linear Mapping ◽  
Finite Dimensional ◽  
Method Of Lagrange Multipliers

The method of Lagrange multipliers for solving a constrained stationary-value problem is generalized to allow the functions to take values in arbitrary Banach spaces (over the real field). The set of Lagrange multipliers in a finite-dimensional problem is shown to be replaced by a continuous linear mapping between the relevant Banach spaces. This theorem is applied to a calculus of variations problem, where the functional whose stationary value is sought and the constraint functional each take values in Banach spaces. Several generalizations of the Euler-Lagrange equation are obtained.

Conjugate Banach spaces

1957 ◽  
Vol 53 (3) ◽  
pp. 576-580 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Linear Mapping ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Linear Mapping ◽  
Conjugate Space ◽  
Necessary And Sufficient ◽  
Weak Topologies

The purpose of this note is to present two characterizations of conjugate Banach spaces. More precisely, we present two conditions, each necessary and sufficient for a (real or complex) Banach space to be isomorphic to the conjugate space of a Banach space, and two corresponding conditions for to be equivalent to the conjugate space of a Banach space. Other characterizations, in terms of weak topologies, have been given by Alaoglu ((1), Theorem 2:1, p. 256, and Corollary 2:1, p. 257) and Bourbaki ((4), Chap, IV, §5, exerc. 15c, p. 122). Here, by the conjugate space* of a Banach space we mean ((2), p. 188) the space of continuous linear functionals over . Two Banach spaces and are said to be isomorphic if there is a one-one continuous linear mapping of onto (its inverse is necessarily continuous by the inversion theorem ((2), Théorème 5, p. 41; (6), Theorem 2·13·7, Corollary, p. 29)); they are said to be equivalent if there is a norm-preserving linear mapping of onto .((2), p. 180).

Projections in Certain Continuous Function Spaces C(H) and Subspaces of C(H) Isomorphic with C(H)

1962 ◽  
Vol 14 ◽  
pp. 385-401
Author(s):  
David W. Dean
Keyword(s):  
Continuous Function ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Linear Mapping ◽  
Topological Spaces ◽  
Continuous Linear ◽  
Hausdorff Spaces ◽  
Linear Spaces ◽  
Continuous Linear Mapping ◽  
Continuous Inverse

If A and B are sets then A — B = {x| x £A, x ∉ B}. This notation is also used if A and B are linear spaces. If X and Y are Banach spaces an embedding of X into Y is a continuous linear mapping u of X onto a closed subspace of F which is 1 — 1. In this case X is said to be embedded in Y. If |ux| = |x| for every x ∈ X (| … | stands for norm), then u embeds Xisometrically into Y. If u is onto then X and Y are isomorphic and if, in addition, |ux| = |x| for every x ∈ X, then X and Y are isometric. Then an embedding u has a continuous inverse u-1 (4, p. 36) defined on uX and this fact is used below without further reference. The conjugate space of X is denoted by X′. Unless otherwise noted, all topological spaces considered are Hausdorff spaces.

Absolutely divergent series and classes of Banach spaces

1983 ◽  
Vol 94 (2) ◽  
pp. 281-289 ◽  
Author(s):  
William H. Ruckle
Keyword(s):  
Normed Space ◽  
Product Space ◽  
Dimensional Space ◽  
Linear Mapping ◽  
Divergent Series ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Continuous Linear ◽  
Continuous Linear Mapping ◽  
Finite Dimensional

A normed space E is said to be series immersed in a Banach space X if for every absolutely divergent series nxn in E there is a continuous linear mapping T from E into X such that nTxn diverges absolutely. The theorem of Dvoretzky and Rogers(1) implies that a normed space E is series immersed in a finite dimensional space if and only if E itself is finite dimensional. In (4) and (9) it was shown that E is series immersed in lp 1 p < if and only if E is isomorphic to a subspace of Lp() for some measure . In particular, E is isomorphic to an inner product space if and only if it is series immersed in a Hilbert space. The property of series immersion was further studied in the papers (7) and (8). The main results in these two papers are conditions on X under which E series immersed in X would imply E locally immersed in X, a condition slightly stronger (formally) than E finitely representable in X.

scholarly journals The pseudodifferential operatorA(x,D)

2004 ◽  
Vol 2004 (8) ◽  
pp. 407-419
Author(s):  
R. S. Pathak ◽  
S. Pathak
Keyword(s):  
Integral Representation ◽  
Pseudodifferential Operator ◽  
Linear Mapping ◽  
Bessel Operator ◽  
Continuous Linear ◽  
Continuous Linear Mapping ◽  
Zemanian Space ◽  
Norm Continuity ◽  
Hankel Convolution ◽  
Symbol Class

The pseudodifferential operator (p.d.o.)A(x,D), associated with the Bessel operatord2/dx2+(1−4μ2)/4x2, is defined. Symbol classHρ,δmis introduced. It is shown that the p.d.o. associated with a symbol belonging to this class is a continuous linear mapping of the Zemanian spaceHμinto itself. An integral representation of p.d.o. is obtained. Using Hankel convolutionLσ,αp-norm continuity of the p.d.o. is proved.

scholarly journals Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

2019 ◽  
Vol 33 (1) ◽  
pp. 97-105
Author(s):  
Hoger Ghahramani ◽  
Saman Sattari
Keyword(s):  
Identity Element ◽  
Complex Banach Space ◽  
Linear Mapping ◽  
Continuous Linear ◽  
Nest Algebra ◽  
Continuous Linear Mapping ◽  
Nest Algebras ◽  
Left Derivation ◽  
Linear Maps ◽  
Generalized Jordan Left Derivation

AbstractLet Alg 𝒩 be a nest algebra associated with the nest 𝒩 on a (real or complex) Banach space 𝕏. Suppose that there exists a non-trivial idempotent P ∈ Alg 𝒩 with range P (𝕏) ∈ 𝒩, and δ : Alg 𝒩 → Alg 𝒩 is a continuous linear mapping (generalized) left derivable at P, i.e. δ (ab) = aδ (b) + bδ (a) (δ (ab) = aδ(b) + bδ(a) − baδ(I)) for any a, b ∈ Alg 𝒩 with ab = P, where I is the identity element of Alg 𝒩. We show that is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras Alg 𝒩 with the property that δ (P ) = 2Pδ (P ) or δ (P ) = 2P δ (P ) − Pδ (I) for every idempotent P in Alg 𝒩.

scholarly journals A generalization of Lagrange multipliers: Corrigendum

1978 ◽  
Vol 18 (1) ◽  
pp. 159-160
Author(s):  
B.D. Craven
Keyword(s):  
Banach Spaces ◽  
Lagrange Multipliers ◽  
Continuous Linear ◽  
Linear Map ◽  
Linear Maps ◽  
Valid Proof

There is a lacuna in the proof of Lemma 1 of [1]; the projector q is assumed without proof. An alternative, valid proof is as follows.LEMMA 1. Let S, U0, V0 be real Banach spaces; let A : S → U0 and B : S → V0 be continuous linear maps, whose null spaces are N(A) respectively N(B); let N(A) ⊂ N(B) : let A map S onto U0. Then there exists a continuous linear map C : U0 → V0 such that B = C ° A.

On Convergence of Projections in Locally Convex Spaces

1966 ◽  
Vol 9 (1) ◽  
pp. 107-110
Author(s):  
J. E. Simpson
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Linear Mapping ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Continuous Linear Mapping ◽  
Basic Assumptions ◽  
Locally Convex ◽  
Strong Dual ◽  
Convex Spaces

This note is concerned with the extension to locally convex spaces of a theorem of J. Y. Barry [ 1 ]. The basic assumptions are as follows. E is a separated locally convex topological vector space, henceforth assumed to be barreled. E' is its strong dual. For any subset A of E, we denote by w(A) the closure of A in the σ-(E, E')-topology. See [ 2 ] for further information about locally convex spaces. By a projection we shall mean a continuous linear mapping of E into itself which is idempotent.

scholarly journals On weighted inductive limits of spaces of Fréchet-valued continuous functions

1991 ◽  
Vol 50 (2) ◽  
pp. 233-242 ◽  
Author(s):  
Antonio Galbis
Keyword(s):  
Continuous Functions ◽  
Linear Mapping ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Inductive Limits ◽  
Continuous Linear Mapping ◽  
Weighted Inductive Limits

AbstractIn this article we continue the study of weighted inductive limits of spaces of Fréchet-valued continuous functions, concentrating on the problem of projective descriptions and the barrelledness of the corresponding “projective hull”. Our study is related to the work of Vogt on the study of pairs (E, F) of Fréchet spaces such that every continuous linear mapping from E into F is bounded and on the study of the functor Ext1 (E, F) for pairs (E, F) of Fréchet spaces.

scholarly journals LAGRANGIAN METHOD FOR ALGORITHM OPTIMIZATION OF RIBBED THIN PLATES

2018 ◽  
pp. 140-147
Author(s):  
R. P. Moiseenko ◽  
O. O. Kondratenko
Keyword(s):  
Lagrange Multipliers ◽  
Lagrange Equation ◽  
Thin Plates ◽  
Algorithm Optimization ◽  
Lagrange Equations ◽  
Equation Solution ◽  
Optimum Parameters ◽  
Equation Analysis ◽  
Optimality Properties ◽  
Method Of Lagrange Multipliers

The paper presents two iteration algorithms for the equation solution using the method of Lagrange multipliers. It is shown that these iteration algorithms do not converge. For comparison, we use the optimum parameters of a ribbed plate obtained by other methods. The proposed method is based on the specific properties of optimality of ribbed plates formulated as a result of the Lagrange equation analysis. These optimum parameters satisfy each of Lagrange equations. The solution of these equations shows that optimization of ribbed plates is possible only with the use of specific optimality properties.

scholarly journals Function spaces and the Mosco topology

1990 ◽  
Vol 42 (1) ◽  
pp. 7-19 ◽  
Author(s):  
Gerald Beer ◽  
Robert Tamaki
Keyword(s):  
Banach Spaces ◽  
Function Space ◽  
Weak Topology ◽  
Continuous Functions ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Compact Open Topology ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Weakly Continuous

Let X and Y be Banach spaces and let C(X, Y) be the functions from X to Y continuous with respect to the weak topology on X and the strong topology on Y. By the Mosco topology τM on C(X, Y) we mean the supremum of the Fell topologies determined by the weak and strong topologies on X × Y, where functions are identified with their graphs. The function space is Hausdorff if and only if both X and Y are reflexive. Moreover, τM coincides with the stronger compact-open topology on C(X, Y) provided X is reflexive and Y is finite dimensional. We also show convergence in either sense is properly weaker than continuous convergence, even for continuous linear functionals, whenever X is infinite dimensional. For real-valued weakly continuous functions, τM is the supremum of the Mosco epitopology and the Mosco hypotopology if and only if X is reflexive.

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