If sum of LI vectors is Cauchy, do the vectors themselves have to be Cauchy?

Question

Say I have a normed space $X$ and a Cauchy sequence $(x_n) \in X$. We can define $W$ to be an arbitrary subspace of X, with the Hamel Basis $\{e_1, e_2, e_3 ...\}.$

Given that $W$ is a subspace of $X$, we can define a Hamel Basis of $X$ as $$\{e_1, e_2, e_3 ... \}\cup\{e'_1,e'_2,e'_3...\}$$ From this, I think we can define unique sequences $(y_n)$ and $(w_n)$ such that $x_n = y_n + w_n$ for all $n \in \mathbb{N}$, and that $w_n \in W$ and $y_n \in \text{Span}(\{e'_1,e'_2,e'_3...\})$ for all $n$. Is it true then that for all normed spaces $X$, that $(y_n)$ and $(w_n)$ have to be Cauchy? $X$ and $W$ are not necessarily complete and not necessarily of finite dimension.

I think it's true, but in my attempts, I can't think of a way to get to where I need to go without using the assumption that $||y+w||\ge||y||,||w||$ for $y \in \text{Span}(\{e'_1,e'_2,e'_3...\})$ and $w \in W$. I don't think that inequality is true in general.

Either pointers or solutions would both be appreciated.

Answer 1

Consider $c_{00}$, the space of all finitely supported sequences equipped with $\ell^2$-norm and let $W = \operatorname{span}\{e_1\}$ and $Y = \left\{x \in c_{00} : \sum_{k=1}^\infty kx_k = 0\right\}$. Then $c_{00} = W \dotplus Y$. Let $$x_n = \left(1,\frac12, \frac13, \ldots, \frac1n, 0, 0, \ldots\right).$$ Then $(x_n)_n$ is Cauchy in $c_{00}$. The unique decomposition $x_n = y_n + w_n$ with $y_n \in Y$ and $w_n \in W$ is $$w_n = ne_1, \quad y_n = x_n - ne_1.$$ Neither sequence $(y_n)_n$ or $(w_n)_n$ is bounded so in particular they aren't Cauchy.

Answer 2

The answer is no. Let $X=(\Bbb R^2)^\infty$ with the supremum norm inherited from $\Bbb R^2$, and define $W=W_1\times W_2\times\cdots$ where each $W_n$ in the $n$th coordinate is spanned by $(n,1)$. Let $(x_n)\subset X$ be the sequence where all coordinates of $x_n$ equal $(0,0)$ except that the $n$th coordinate equals $(0,\frac1n)$. Then $(w_n)\subset W$ is the sequence where all coordinates of $w_n$ equal $(0,0)$ except that the $n$th coordinate equals $(1,\frac1n)$, while $(y_n)\subset X\setminus W$ is the sequence where all coordinates of $y_n$ equal $(0,0)$ except that the $n$th coordinate equals $(-1,0)$; neither of these sequences will be Cauchy (since the nonzero coordinate keeps changing indices).