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 (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:

    The general who advances without coveting fame and retreats without fearing disgrace, whose only thought is to protect his country and do good service for his sovereign, is the jewel of the kingdom.
    Sun Tzu (6th–5th century B.C.)

    The duty of the State toward the citizen is the duty of the servant to its master.... One of the duties of the State is that of caring for those of its citizens who find themselves the victims of such adverse circumstances as makes them unable to obtain even the necessities for mere existence without the aid of others.... To these unfortunate citizens aid must be extended by government—not as a matter of charity but as a matter of social duty.
    Franklin D. Roosevelt (1882–1945)

    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)