Discontinuous Linear Map - General Existence Theorem

General Existence Theorem

Discontinuous linear maps can be proven to exist more generally even if the space is complete. Let X and Y be normed spaces over the field K where K = R or K = C. Assume that X is infinite-dimensional and Y is not the zero space. We will find a discontinuous linear map f from X to K, which will imply the existence of a discontinuous linear map g from X to Y given by the formula g(x) = f(x)y₀ where y₀ is an arbitrary nonzero vector in Y.

If X is infinite-dimensional, to show the existence of a linear functional which is not continuous then amounts to constructing f which is not bounded. For that, consider a sequence (e_n)_n (n ≥ 1) of linearly independent vectors in X. Define

for each n = 1, 2, ... Complete this sequence of linearly independent vectors to a vector space basis of X, and define T at the other vectors in the basis to be zero. T so defined will extend uniquely to a linear map on X, and since it is clearly not bounded, it is not continuous.

Notice that by using the fact that any set of linearly independent vectors can be completed to a basis, we implicitly used the axiom of choice, which was not needed for the concrete example in the previous section.