A Discussion About Dimension

The basis for $\mathbb {R}^{n}$ given in the text is called the "standard basis." It is a basis, but not the only basis, for $\mathbb {R}^{n}$. Give another basis for $\mathbb {R}^{3}$, and show that it is in fact a basis.

Prove that $\ell^{1}$ is infinite-dimensional. Explain why your proof also shows that $\ell^{\infty }$ is infinite-dimensional.

Prove that $C([0,1])$ is infinite-dimensional.