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Mathematics for Physics

247 Dual Space

A set of linear transformations from a domain of vectors to a codomain of scalars that is used to associate each vector with a scalar value.

Note: Also called dual vector space. Also called algebraic dual space. Also called conjugate space. Also called adjoint space.

definition [d] (Dual Space = Dual Vector Space = Algebraic Dual Space = Conjugate Space = Adjoint Space) The vector space \(V^{*}\) of all linear functionals on a vector space \(V\):

  • \(V^{*} = \{\, \varphi : V \rightarrow K \mid \varphi \text{ is linear} \,\}\) .

where

  • \(V\) is a vector space over a field \(K\).
  • \(K\) is the scalar field.
  • \(\varphi\) is a linear functional from \(V\) to \(K\).
  • \(V^{*}\) is the dual space.
  • elements of \(V^{*}\) are called covectors.

Note:

  • \(K\) is typically \(\mathbb{R}\).
  • \(K\) is also typically \(\mathbb{C}\).
  • \(V^{*}\) is also written \(V'\).
  • a linear functional \(\varphi\) maps each vector to a scalar in \(K\).
  • if \(\dim V = n < \infty\), then \(\dim V^{*} = n\).

247.1 Elementary Example

247.1.1 Simple

The dual space is the set of all linear functionals on \(V\). On \(\mathbb{R}^{2}\), each functional is a row of two components.

\[ V = \mathbb{R}^{2} \]

\[ \varphi(x_{1},x_{2}) = 3 x_{1} - x_{2} \]

\[ \varphi \in V^{*} \]

where

  • \(V^{*}\) is the dual space.
  • \(\varphi : V \rightarrow \mathbb{R}\) is a linear functional.

247.1.2 General

If \(\dim V = 3\), then \(\dim V^{*} = 3\), with dual basis \(e^{1}, e^{2}, e^{3}\).

\[ V = \mathbb{R}^{3},\quad V^{*} = \operatorname{span}\{ e^{1},\ e^{2},\ e^{3} \} \]

\[ e^{i}(e_{j}) = \delta^{i}_{\ j} \]

where

  • \(e^{i}\) are the dual basis functionals.
  • \(\delta^{i}_{\ j}\) equals \(1\) if \(i = j\) and \(0\) otherwise.

247.2 References

  1. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — algebraic dual space; also adjoint space, conjugate space.
  2. Dummit, D. S., & Foote, R. M. Abstract Algebra. — dual space, dual vector space, algebraic dual space.
  3. Szekeres, P. A Course in Modern Mathematical Physics. Cambridge University Press, 2004. — dual space, covectors, 1-forms.