Let V Be the Vector Space of Continuous Fzunctions F R R Show That the Product Defined Below
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Vector Spaces
Recall the theorem about sums and multiples of continuous functions: If and are continuous on an interval and is any real number, then
and | ||
This theorem says that the set of functions continuous on is closed under sums and constant multiples. Thus we have a situation which may yield a vector space. With this in mind, we define as the set of functions continuous on along with addition of functions and constant multiplication of functions defined in the usual way. To see whether is a vector space, we must check the vector space axioms. Closure holds by the theorem stated above. The properties
are easily checked. The zero vector is the constant function (it is continuous) which is identically zero on . Given , the function is continuous and is the additive inverse of . Therefore, is a vector space.
Differentiable functions are another important set of functions in calculus. If we define as the set of functions defined on with (that is, is continuous) and give the same operations as , then is a subspace of . To see this, we need only check closure. Since a theorem from calculus tells us that the sum and constant multiples of differentiable functions are differentiable, we have the necessary closure. So is a vector space in its own right as well as a subspace of .
Example 1 Give an example of a function in which is not in . This shows that is a proper subspace of .
Solution A continuous function which is not differentiable will be sufficient. The function is continuous on but fails to have a derivative at .
Several functions from calculus are differentiable an infinite number of times. For example, , , , and all polynomials are infinitely differentiable over all . Thus we can define vector spaces
and even
Example 2 Give a example of a function which is in but is not in .
Solution Let . Then and . Although is continuous on is not, because is undefined.
Examples 1 and 2 illustrate the subspace relationship
among these vector spaces.
The vector space differs from the vector spaces we have been studying in that is not finite-dimensional. To see this, suppose that . Then a finite basis of continuous functions in would exist, and any set of functions would have to be linearly dependent. However, is a set of functions in which are linearly independent. So it is impossible to have for any . Since polynomials are infinitely differentiable, the same argument shows that is not finite-dimensional for any .
Although is not finite-dimensional, there are important finite-dimensional subspaces of .
Example 3 Show that the set of all solutions to the equation3.3
is a one-dimensional subspace of
Solution We recall that the functions3.4 satisfying are , where is arbitrary. That is, . To show that is a subspace of , we first note that since is a continuous function, . Now closure of under the operations must be shown. Since and , we have the closure, so is a subspace of . A basis for is . Thus .
Example 4 Show that the set of all solutions to the equation
is a two-dimensional subspace of
.
Solution The solutions of the equation are of the form . Showing that is a subspace is done in the same way as Example 3. A basis for is . That spans is easy. For the linear independence consider . The equation must hold for all . Substitute to find . Substitute to obtain .
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Source: https://www.math.tamu.edu/~dallen/DistanceEd/Math640/chapter3/node20.html
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