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Chapter 4: Q4.5-28Q (page 191)

Show that the space \(C\left( \mathbb{R} \right)\) of all continuous functions defined on the real line is infinite-dimensional.

Short Answer

Expert verified

It is proved that the space \(C\left( \mathbb{R} \right)\) of all continuous functions is infinite-dimensional.

Step by step solution

01

Explain finite-dimensional

Suppose \(H\) is a subspace of a finite-dimensional vector space \(V\). Then, according totheorem 11,anylinearly independent setin \(H\) can be expanded, if necessary, to a basis for \(H\). Also, \(H\) is finite-dimensional and \(\dim H \le \dim V\).

02

Show that the space \(C\left( \mathbb{R} \right)\) of all continuous functions is infinite-dimensional

The space \(C\left( \mathbb{R} \right)\) is the subspace of \({\mathop{\rm P}\nolimits} \). According to theorem 11, if the space \(C\left( \mathbb{R} \right)\) is finite-dimensional, then \({\mathop{\rm P}\nolimits} \) must also be finite-dimensional. \(C\left( \mathbb{R} \right)\) must also be infinite-dimensional because \({\mathop{\rm P}\nolimits} \) is infinite-dimensional, according to Exercise 27.

Thus, it is proved that the space \(C\left( \mathbb{R} \right)\) of all continuous functions is infinite-dimensional.

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Most popular questions from this chapter

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

14. Show that if \(Q\) is an invertible, then \({\mathop{\rm rank}\nolimits} AQ = {\mathop{\rm rank}\nolimits} A\). (Hint: Use Exercise 13 to study \({\mathop{\rm rank}\nolimits} {\left( {AQ} \right)^T}\).)

In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.

\({\bf{1}} - {\bf{2}}{t^{\bf{2}}} - {t^{\bf{3}}}\), \(t + {\bf{2}}{t^{\bf{3}}}\), \({\bf{1}} + t - {\bf{2}}{t^{\bf{2}}}\)

Given \(T:V \to W\) as in Exercise 35, and given a subspace \(Z\) of \(W\), let \(U\) be the set of all \({\mathop{\rm x}\nolimits} \) in \(V\) such that \(T\left( {\mathop{\rm x}\nolimits} \right)\) is in \(Z\). Show that \(U\) is a subspace of \(V\).

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.

a. What is the dimension of range of T if T is one-to-one mapping? Explain.

b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.

Suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss.
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