Dual Space

The dual space of a vector space E is the whole of the linear forms on E. the structure of a space and that of its dual is very dependant. The end of this article has some results on the bonds between dual space and hyperplanes, which allows a “geometrical” comprehension of certain properties of the linear forms.

The topological Dual is an alternative very considered in functional analysis, when the vector space is provided with an additional structure of topological vector Space.

Definitions

Are (K, +, X) a body, E a vectorial K-space.

One calls linear Forme on E all Linear application of E towards K, i.e. any application \ phi: E \ to \ mathbb {K} \, \! such as:

\ forall (X, there) \ in E^2, \ forall \ lambda \ in \ mathbb {K}, \ phi (\ lambda X + there) = \ lambda \ phi (X) + \ phi (there) \, \!

The unit \ \ mathcal {L} (E, \, \ mathbb {K}) of the linear forms on E is a K-space vectorial, known as dual space of E; it is noted E^* \, \! .

If \ phi \, \! is an element of E^* \, \! and x \, \! an element of E \, \! , one writes sometimes \ langle \ phi, X \ rangle \, \! for \ phi (X) \, \! . This notation is known as hook of duality .

Examples

Case of a space préhilbertien.

If the vector space E is a Espace préhilbertien, i.e. provided with a scalar Produit, there is an average naturalness “to plunge” E \, \! in E^* \! , i.e. to associate with each element of E an element of dual, and this so as to form an isomorphism between E \, \! and a subspace of E^* \! : with each element x \, \! of E one associates the linear form \ phi_x: E \ to \ mathbb {K}, there \ mapsto \ langle X, there \ rangle \, \! . Then the f application: E \ to E^*, X \ mapsto \ phi_x \, \! is a Linear application injective, therefore space E is isomorphous with the subspace f (E) \, \! of E^* \, \! .

Duality in finished dimension

If space E is of finished size n \, \! , then dual space E^* \, \! , isomorphous with E, is him also of dimension n \, \! .

Obtaining this result by basic construction “dual”:

If \ mathcal {B} = (e_i) _ {I \ in I} \, \! is a base, one can define the forms Coordonnée S : for each i \ in I \, \! the coordinated form e_i^* \, \! associates with each vector of E its i-ième coordinate in base \ mathcal {B} \, \! .

\ forall X \ in E \, \! one writes x = \ sum_ {J \ in I} x_j e_j \, \! and then e_i^* (X) = x_i \, \! .

Theorem: \ mathcal {B} ^* = (e_i^*) _ {I \ in I} \, \! is a base of E* known as bases dual base \ mathcal {B} \, \! , and any linear form \ phi \, \! on E, is written then: \ phi = \ sum_ {I \ in I} \ phi (e_i) e_i^* \, \!

Since \ mathcal {B} ^* \, \! is a base of E^* \, \! , one from of deduced the result announced higher:

\ dim E = \ dim E^* \, \! . In finished dimension, a space thus has same dimension as its dual space. Let us notice that one cannot affirm in the general case that a vector space is isomorphous with its dual: this is false for certain vector spaces of infinite size.

Example

The polynomials of Lagrange associated with scalars x_0, x_1, \ dowries, x_n (see Lagrangian Interpolation) form a base of the whole of the polynomials whose dual base is formed of the functions of evaluations \ \ phi_i (P) =P (x_i) .

Orthogonal

E is an unspecified vector space (one does not suppose of a finished size).

If has is a subspace of E \, , one defines the orthogonal one of has in E^* \, by:

A^ \ circ= \ {\ phi \ in E^* \ colonist \ forall X \ in has, \ langle \ phi, X \ rangle=0 \} \,

If B is a subspace of E^* \, , one defines orthogonal B in E \, by:

B^ \ bot= \ {X \ in E \ colonist \ forall \ phi \ in B, \ langle \ phi, X \ rangle=0 \} \,

One should not confuse the notion of orthogonal of a subspace in the theory of duality with orthogonality in the theory of the Euclidean spaces.

Representation of the subspaces

This paragraph presents a very important application of the study of dual space: the representation of a subspace like intersection of Hyperplan S. One is restricted here with the case of a vector space of finished size.

Tally: E is a vectorial K-space of finished size N.

That is to say F a subspace of dimension p (distinct from E); there is thus p < N.

Then, there exists Q = N - p independent linear forms \ phi_1,…, \ phi_q \, \! such as:

F= \ bigcap_ {i=1} ^q \ ker \ phi_i \, \! i.e. \ forall X \ in E, (X \ in F \ Leftrightarrow \ phi_1 (X) =0,…, \ phi_q (X) =0) \, \!

This theorem generalizes the known elementary results in dimension 2 or 3 on the representation of right-hand side or plan by equations. In particular, in a vector space of dimension 3, the intersection of 2 independent plans is a line.

Foot-note: one should not confuse the concept of right-hand side or of plan in a space refines (which corresponds to the geometrical intuition) and that, used here, vectorial line or of vectorial plan. One calls vectorial right-hand side a subspace of dimension 1, and vectorial plan a subspace of dimension 2.

One can thus represent a subspace F of dimension p by Q independent linear equations, where Q = \ dim E p \, \! .

See too

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