Question

Prove that the space IR[0,1] is infinite-dimensional.

Answer 1

The space IR[0,1] is infinite-dimensional because there exist an infinite number of linearly independent vectors in this space, such as a set of polynomials with increasing degrees.

Step-by-step explanation:

To prove that the space IR[0,1] is infinite-dimensional, we can show that there exist an infinite number of linearly independent vectors in this space. One way to do this is to consider a set of polynomials with increasing degrees.

For example, we can start with the constant polynomial 1, then add the linear polynomial x, then add the quadratic polynomial x^2, and so on. These polynomials form a basis for the space IR[0,1], and since there are infinitely many degrees, the space is infinite-dimensional.