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Chapter 4: Q55E (page 177)

Show that the space F(R,R)of all function from R to R is infinite dimensional.

Short Answer

Expert verified

The solutionFR,Ris infinite dimensional.

Step by step solution

01

Explanation of the solution

Assume that the opposite of F(R,R) is n-dimensional for some n.

The n+1 polynomials as follows.

1,x,x2,...,xn鈭�F(R,R)are linearly independent.

02

Draw the conclusion

This contradicts the fact that n is the largest possible dimension for an n-dimensional space that m=n.

Therefore, the assumption is wrong.

Thus,F(R,R)is infinite dimensional.

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