How to prove that sequence spaces $l^{p}, l^{\infty}$ and function space $C[a. b]$ are of infinite dimension

Question

I am studying about the sequence space $l^{p}, l^{\infty}$ and function space $C[a. b]$. It is mentioned in the book that all of these spaces are of infinite dimension. I want to prove that these spaces are of infinite dimension. I am not finding a way to proceed.

Any help and suggestions would be helpful to me.

Thanks

Answer 1

You just need to construct infinitely many linearly independent vectors. For $l_p$ and $l_{\infty}$ take vectors $(0,0,0,...0,1.....).$ For $C[a,b]$ polynomials $x^n$ will do.

Answer 2

You can exhibit an infinite linearly independent family:

• For $\ell^p$ or $\ell^{\infty}$, consider the sequences of the form $(0,\dots,0,1,0, \dots)$.

• For $C([a,b])$, consider the functions $t \mapsto e^{ct}$.

Remark: In fact, the second point shows that $C([a,b])$ has dimension $\mathfrak{c}$.

Answer 3

Hint: You need to find infinite linearly independent familiy in each of these spaces. I suggest you to search them among functions with disjoint supports.