Examples of Dual Spaces Note. Orthogonality 7 1.8. Dual Spaces Definition 1 (Dual Space) Let V be a finite dimensional vector space. It follows from this discussion that the dual space any function space in which the test functions are densely included (e.g. Lp SPACES Let V be a vector space and U ⊂V.IfU is closed under vector addition and scalar multiplication, then U is a subspace of V. Proof. 1.2. Corollary. The set of all linear maps fL: V ! H0m U,Hm Rn) can be identified with a subspace of the space of distributions. Theorem 1.1.1. The dual space of a Banach space consists of all bounded linear functionals on the space. Duality 6.1. The dual space 2 1.3. Let V be any real or complex vector space with a norm kvk again, and let us check that the corresponding dual space V∗ is complete with respect to the dual norm kλk∗. The dual of a vector space 1.1. dimCC= 1, dimRC= 2, dimQR= 1. Linear functionals as covectors; change of basis 4 1.5. This operation T is also commonly known as the adjoint. Contents There are two types of dual spaces: the algebraic dual space, and the continuous dual space. The notation Jj!i is a bit clumsy, even if its meaning is clear, and Dirac’s h!j, called a \bra", provides a simpler way to denote the same object, De nition 7.12. Recall that the dual space of a normed linear space X is the space of all bounded linear functionals from X to the scalar field F, originally denoted B(X,F), but more often denoted X∗. The transpose of a linear transformation 8 1. The bidual 7 1.9. Application: interpolation of sampled data 5 1.7. 3.2. Remark. Let V be a vector space of dimension nwith dual space V . 6.1. Deflne the dimension of a vector space V over Fas dimFV = n if V is isomorphic to Fn. We remark that this result provides a “short cut” to proving that a particular subset of a vector space is in fact a … In this section we find the duals of the `p spaces for The algebraic dual space is defined for all vector spaces. The dual space of a vector space V is de ned to be the space of all linear functions v : V !R. If a monomial is a product of pelements of V with qelements of V , then the Dual space Definition – Dual space Suppose X is a normed vector space over R. Its dual X∗ is then the set of all bounded linear operators T :X → R, namely X∗ =L(X,R). With this de nition we have the following action, if v2V and v 2V then we have That is, if the inclusion, C0 U X has a dense image, then X D U is a continuous injection. When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space. Functionals and hyperplanes 4 1.6. Dual basis 3 1.4. Now if T 2L(V;W), we can de ne the dual transformation T , by T : W !V . Examples of Dual Spaces 1 Chapter 6. A 1-form is a linear transfor- mation from the n-dimensional vector space V to the real numbers. (a) A linear functional on V is a function ~u∗: V → IR that is linear in the sense that ~u∗(~v + w~) = ~u∗(~v) +~u∗(w~) and ~u∗(α~v) = α~u∗(~v) for all ~u,w~ ∈ V and all α ∈ IR. Chapter 1 Forms 1.1 The dual space The objects that are dual to vectors are 1-forms. The 1-forms also form a vector space V∗ of dimension n, often called the dual space of the original space V of vectors. Wg over Fis homomorphism, and is denoted by homF(V;W). The full tensor algebra of V is the sub-algebra of the tensor algebra T(V V ) generated by monomials v i 1 v i 2:::v i k such that each v i belongs either to V or to V . LINEAR FUNCTIONALS AND THE DUAL SPACE 25 a label on J and, for example, write it as Jj!i j i. Linear functionals. If Xis a real Banach space, the dual space of X consists of all bounded linear functionals F: X!R, with norm kFk X = sup x2Xnf0g jF(x)j kxk X <1: 84 7.