dual space
The dual space collects every continuous way of turning vectors into numbers. If a vector space holds the objects of interest, its dual holds the measurements you can make of them — each measurement linear and continuous. Studying a space through its dual is like studying a shape through all its shadows: often the shadows are easier to handle and reveal everything that matters.
For a normed space X, the dual space X* is the set of all bounded linear functionals f : X -> scalars, made into a normed space by the operator norm ||f|| = sup over ||x|| <= 1 of |f(x)|. A fundamental and reassuring fact: X* is always complete — a Banach space — even when X itself is not, because the scalar field is complete. So duals are automatically among the best-behaved spaces.
Different concrete spaces have different duals, and identifying them is a recurring theme. The dual of l^p is l^q with 1/p + 1/q = 1 for 1 <= p < infinity; the dual of a Hilbert space is (a copy of) itself by Riesz; the dual of L^p is L^q similarly. One must be careful: the dual of l^infinity is strictly larger than l^1, so the pattern does not extend to the endpoint, a reminder to check hypotheses rather than extrapolate.
The dual of l^1 is l^infinity: a bounded functional on l^1 has the form f(x) = sum x_n y_n for a unique bounded sequence y = (y_n), and ||f|| = sup_n |y_n| = ||y||_infinity. For instance y = (1, -1, 1, -1, ...) gives the functional f(x) = x_1 - x_2 + x_3 - ... of norm exactly 1.
Identifying the dual of l^1 with l^infinity, with a concrete unit-norm functional.
Mapping X into its second dual (X*)* by x -> (evaluation at x) is always an isometry; when it is onto, X is called reflexive. Hilbert spaces and L^p for 1 < p < infinity are reflexive, but L^1, L^infinity, and C[0, 1] are not — a structural distinction with real consequences for weak convergence.