linear functional
A linear functional is the simplest kind of measurement on a vector space: it eats a vector and returns a single number, doing so in a way that respects addition and scaling. Averaging a function, reading off one coordinate, integrating against a fixed weight — all of these are linear functionals. They are the bricks from which the dual space and the whole machinery of duality are built.
Formally, a linear functional on a vector space V over a field of scalars is a map f : V -> scalars with f(a x + b y) = a f(x) + b f(y) for all vectors x, y and scalars a, b. When V is normed, f is called bounded, or equivalently continuous, if |f(x)| <= M ||x|| for some constant M; the least such M is the norm ||f||, and the bounded functionals are precisely the members of the dual space V*.
In finite dimensions every linear functional is automatically continuous, so the distinction is vacuous there. In infinite dimensions it is not: discontinuous linear functionals exist (their construction requires a Hamel basis and the axiom of choice), and they are invisible to direct construction. Geometrically, the level set f(x) = 0 is a hyperplane, and f is continuous exactly when that hyperplane is closed rather than dense.
On C[0, 1] the evaluation functional f -> f(1/2) is linear and bounded: |f(1/2)| <= max|f| = ||f||, so ||evaluation|| = 1 (take f = 1). The definite integral f -> integral from 0 to 1 of f is another bounded functional, also of norm 1. Both live in the dual of C[0, 1].
Evaluation at a point and definite integration: two bounded functionals on C[0, 1], each of norm 1.
A linear functional is completely determined by its kernel together with its value at one vector off the kernel; two functionals with the same kernel are scalar multiples of each other. This is why continuity of f reduces to a single geometric question about the closedness of one hyperplane.