Definition:Linear Functional

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Definition

Let $E$ be a vector space over a field $\GF$.

Let $D$ be a linear subspace of $E$.

A mapping $f : D \to \GF$ is called a linear functional if and only if:

$\map f {\alpha x + \beta y} = \alpha \map f x + \beta \map f y$

holds for all $x, y$ in $L$ and for all $\alpha, \beta$ in $\GF$.

Notation

It is customary in functional analysis to omit the parentheses around the argument of linear functionals whenever this can be done unambiguously and without loss of clarity.

For example, one commonly writes $\map f {\alpha x + y} = \alpha f x + f y$ to denote the linearity of $f$ as described above.

This is supposedly done to allow the lazy to avoid the tedious writing of parentheses, and has the dubious aim of making formulae look more appealing.

It is therefore important to strongly keep in mind which letters denote scalars, functionals and elements of the vector space, so as to avoid confusion.

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra