[Math] infinite-dimensional Banach spaces has linear subspaces of finite-codimension that are not closed

Question

I want to show that

In every infinite-dimensional Banach spaces there are linear subspaces of finite-codimension that are not closed .

There is a hint for this question that says use Zorn lemma.

Can someone help.

Answer 1

I don't know if this is exactly what David had in mind, but here is one construction:

Given a Hamel basis $\{e_\alpha\}$ you can make any vector space $X$ a normed space (norm properties can be checked) by setting $$ x = \max{\{|c_1|,\ldots,|c_k|\}} $$ where $x = c_1e_1 + \cdots + c_ke_k$ is the unique expression of $x$ as a finite linear combination of basis elements.

Fix some basis element $e_\alpha$. Then define a linear functional $\phi$ on $X$ by $$ \phi(e_\alpha) = 1 $$ $$\phi(e_\beta) = 0 $$ for all $\beta \neq \alpha$. The codimension of the kernel of $\phi$ is 1. However the kernel is not closed if $X$ is infinite-dimensional, as can be seen by taking any sequence of basis elements $e_1,e_2,\ldots$ all distinct, and distinct from $e_\alpha$, noting that they cannot converge to any element in $X$.