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 (en)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.
Read more about this topic: Discontinuous Linear Map
Famous quotes containing the words general, existence and/or theorem:
“He who never sacrificed a present to a future good or a personal to a general one can speak of happiness only as the blind do of colors.”
—Olympia Brown (1835–1900)
“Creation destroys as it goes, throws down one tree for the rise of another. But ideal mankind would abolish death, multiply itself million upon million, rear up city upon city, save every parasite alive, until the accumulation of mere existence is swollen to a horror.”
—D.H. (David Herbert)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)