Proving $l^1$ and $l^\infty$ are infinite dimensional

Question

How to prove $l^1$ and $l^\infty$ are infinite dimensional spaces.

I know that a space is infinite dimensional if it has a subspace with infinite dimensions. But I don't know how to proceed. Any help will be appreciated.

Answer 1

In both spaces You have an infinite linearly independent set: $(1,0,0,...),(0,1,0,...), (0,0,1,...)$.

Answer 2

Suppose $l^1$ (or $l^\infty$) is finite dimensional, and let $\{e_n\}_1^N$ be a basis. Consider $\{0_n\}_1^{N+1}$ where $0_n$ is the vector zero everywhere and $1$ in the $n$-th position. You now get a contradiction as the set $\{0_n\}_1^{N+1}$ is linearly independent, implying that the cardinality of $\{e_n\}_1^N$ is not the smallest possible for collections of linearly independent vectors.