I have a few questions about the following proof taken from https://math.berkeley.edu/~sarason/Class_Webpages/solutions_202B_assign11.pdf.
Prove that the closed unit ball of an infinite-dimensional Banach space is not compact.
Proof. Let $B$ be an infinite-dimensional Banach space. Choose any unit vector $x_1\in B$, and let $A_1$ be the subspace spanned by $x_1$. In the quotient space $B/A_1$, choose a coset of norm $\frac{1}{2}$ and then a representative $x_2$ of that coset of norm at most 1. Then $\|x_2−x_1\| \geq \frac{1}{2}$. Let $A_2$ be the subspace spanned by $x_1$ and $x_2$,and note that $A_2$ is closed. In the quotient space $B/A_2$, choose a coset of norm $\frac{1}{2}$ and then a representative $x_3$ of that coset of norm at most 1. Then $\|x_3−x_2\| \geq \frac{1}{2}$ and $\|x_3−x_1 \| \geq \frac{1}{2}$. Continuing in this way, we obtain a sequence $\{x_n\}$ of vectors in the closed unit ball of $B$ such that $\|x_m−x_n\| \geq \frac{1}{2}$ whenever $m\neq n$. The sequence $\{x_n\}$ then has no convergent subsequences, implying that the closed unit ball in $B$ is not compact.
My questions are:
1) How do we know that a coset with a norm $\frac{1}{2}$ exists in $B/A_1$?
2) Why does the closedness of $A_2$ matter?
3) Why does it matter if $B$ is a Banach space?
Best Answer
You may want to review the definition and properties of quotients of a normed space.
Because $B / A_1$ is a normed space, and it is not the zero space unless $A_1 = B$, which is not the case here ($B$ is infinite-dimensional and $A_1$ is one-dimensional). So it contains at least one nonzero element (remember elements of $B / A_1$ are cosets), and you can multiply this by an appropriate scalar to obtain something of norm $1/2$.
If you take a quotient of a normed space by a non-closed subspace, in general you only get a seminormed space (the resulting "norm" may fail to be positive definite). Since we want $B / A_2$ to actually be a normed space (see step 1), we need $A_2$ to be closed.
It doesn't matter. It is also true that "the closed unit ball of an infinite-dimensional normed space is not compact".