linear functional
A linear functional is a linear map that eats a vector and spits out a single scalar: f : V -> F, where F is the field (think R or C). It is the simplest possible measurement you can make on a vector that respects the vector-space structure, meaning f(a*u + b*v) = a*f(u) + b*f(v) for all scalars a, b and vectors u, v.
Concretely, once you fix coordinates every functional is just a weighted sum of the entries: f(x) = c_1*x_1 + ... + c_n*x_n. So a functional is a row of numbers waiting to be paired against a column of numbers. Examples: 'read off the third coordinate', 'sum the entries', 'evaluate this polynomial at 0', or the integral that sends a function to its average value.
Functionals are the atoms of duality. Every richer object here — dual space, dual basis, transpose, annihilator — is built out of them. The key mental shift from a first course is to treat 'measuring a vector' as itself a thing that lives in its own vector space, on equal footing with the vectors being measured.
Caveat: in infinite dimensions not every linear map V -> F is continuous, and analysts usually want only the continuous ones (the topological dual). In finite dimensions this distinction evaporates — every linear functional is automatically continuous — so here 'functional' just means 'linear map to scalars'.
Every linear functional on R^n is a fixed row c^T paired against the input column x.
On R^n a functional is exactly a row vector c, acting by f(x) = c*x. The level sets f(x) = constant are parallel hyperplanes, and c points across them — that geometric picture is the bridge to gradients and dual vectors.