Discontinuous Linear Map - A Nonconstructive Example

A Nonconstructive Example

An algebraic basis for the real numbers as a vector space over the rationals is known as a Hamel basis (note that some authors use this term in a broader sense to mean an algebraic basis of any vector space). Note that any two noncommensurable numbers, say 1 and π, are linearly independent. One may find a Hamel basis containing them, and define a map f from R to R so that f(π) = 0, f acts as the identity on the rest of the Hamel basis, and extend to all of R by linearity. Let {r_n}_n be any sequence of rationals which converges to π. Then lim_n f(r_n) = π, but f(π) = 0. By construction, f is linear over Q (not over R), but not continuous. Note that f is also not measurable; an additive real function is linear if and only if it is measurable, so for every such function there is a Vitali set. The construction of f relies on the axiom of choice.

This example can be extended into a general theorem about the existence of discontinuous linear maps on any infinite-dimensional normed space (as long as the codomain is not trivial).