Linear Functionals

Definition

A linear functional on a vector space $V$ over a field $\F $ is a linear function from $V$ to $\F $. In other words, a linear function is an element of $\mathcal{L} (V, \F )$.

Notation

We tend to denote linear functionals by $\phi : V \to \F $, a mnemonic for functional.

Examples

1. Define $\phi : \R ^3 \to \R $ by

   \[ \phi (x, y, z) = 4x - 5y + 2z \]

   $\phi $ is a linear function on $\R ^3$
2. Define $\phi : \C ^n \to \C $ by

   \[ \phi (x_1, \dots , x_n) = c_1x_1 + c_2x_2 + \cdots + c_nx_n \]

   where $c_1, \dots , c_n \in \C $. $\phi $ is a linear functional on $\C ^n$.
3. Let $(c_n)_{n \in \N } \in \ell ^{\infty}$. Define $F_c: \ell ^1 \to \C $ by

   \[ F_c((x_n)_{n \in \N }) = \sum_{n = 1}^{\infty} c_n x_n. \]

4. As usual, denote the set of real polynomials by $\mathcal{P} (\R )$. Define $\phi : \mathcal{P} (\R ) \to \R $ by

   \[ \phi (p) = 3p''(5) + 7p(4) \]

   $\phi $ is a linear functional on $\mathcal{P} (\R )$.
5. As usual, denote the set of real polynomials by $\mathcal{P} (\R )$. Define $\phi : \mathcal{P} (\R ) \to \R $ by

   \[ \phi (p) = \int _{[0,1]} p \]

   $\phi $ is a linear functional on $\mathcal{P} (\R )$.