Applications of Functional Analysis in EngineeringFunctional analysis owes its OrIgms to the discovery of certain striking analogies between apparently distinct disciplines of mathematics such as analysis, algebra, and geometry. At the turn of the nineteenth century, a number of observations, made sporadically over the preceding years, began to inspire systematic investigations into the common features of these three disciplines, which have developed rather independently of each other for so long. It was found that many concepts of this triad-analysis, algebra, geometry-could be incorporated into a single, but considerably more abstract, new discipline which came to be called functional analysis. In this way, many aspects of analysis and algebra acquired unexpected and pro found geometric meaning, while geometric methods inspired new lines of approach in analysis and algebra. A first significant step toward the unification and generalization of algebra, analysis, and geometry was taken by Hilbert in 1906, who studied the collection, later called 1 , composed of infinite sequences x = Xb X 2, ... , 2 X , ... , of numbers satisfying the condition that the sum Ik"= 1 X 2 converges. k k The collection 12 became a prototype of the class of collections known today as Hilbert spaces. |
Contents
Physical Space Abstract Spaces | 1 |
Basic Vector Algebra | 13 |
Minkowski Inequality | 23 |
Copyright | |
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a₁ approximate solution arbitrary Axiom basis Bessel Bessel's inequality boundary conditions boundary value problem Cauchy-Schwarz inequality Chapter coefficients Compare components consider convergence coordinate coordinate vectors corresponding defined definition denote derivative dimension Dirichlet displacement distance dx dy eigenvalues elastic elements equilibrium Euclidean space exact solution example Figure finite function f function space function vectors geometric given h₁ h₂ Hilbert space hypercircle hyperplane hypersphere Illustration implies infinite-dimensional inner product inner product space integral intersection linear manifold linearly independent lower bound method metric norm notation operator orthogonal projection perpendicular plane position vector preceding equation Rayleigh-Ritz real numbers represented respectively S₁ S₂ satisfies scalar sequence strain energy stress symmetry theorem tion torsion translated subspaces triangle triangle inequality u₁ upper bound V₁ vector f vector space Y₁ zero vector ΤΩ