Is there a norm in which the vector space of all sequences with the induced metric is complete?

Question

The question of knowing whether is possible put a norm in a vector space is something new to me. I liked the Ivo ideas and I have wondered a little more about a particular case.

Is there a norm in which the vector space of all sequences with the induced metric is complete? This question sounds harder to answer. The completion of a metric space is something that is usually done to achieve this result. Is the completion of the norm given by Ivo ideas isomorphic to the vector space of all sequences? If not, is there a norm in which the vector space of all sequences with the induced metric is complete?

Answer 1

By "Ivo ideas" I presume you mean, given a Hamel basis $B$ of vector space $V$, the supremum norm $\|v\| = \sup_{\beta \in B} |a_\beta|$ where $v = \sum_{\beta \in B} a_\beta \beta$.

An infinite-dimensional vector space $V$ with this norm is not complete. For example, let $\beta_i$, $i = 1,2,3,\ldots$, be a sequence of distinct members of $\beta$, and consider the sequence $v_n = \sum_{j=1}^n 2^{-j} \beta_{j}$. This is a Cauchy sequence, but it does not have a limit in this norm.

On the other hand, given a vector space $V$, to construct a norm in which it is complete all you need is to find a Banach space $B$ (i.e a complete normed linear space) with the same dimension (i.e. a Hamel basis of the same cardinality as that of $V$). Then the two spaces are isomorphic as vector spaces, and you just carry the norm of $B$ over to $V$. Specifically, if $\| \cdot \|_B$ is the norm of $B$, $(\alpha_i)_{i \in I}$ a Hamel basis of $V$ and $(\beta_i)_{i \in I}$ a Hamel basis of $B$ with the same index set, you define $$ \|v\|_V = \|\phi(v)\|_B$$ where if $v = \sum_i c_i \alpha_i$ is the representation for $v$ in the Hamel basis, $\phi(v) = \sum_i c_i \beta_i$.

Answer 2

To help people out, any separable Banach space have Hamel dimension of $\mathfrak{c}$, the second Beth number, as proven here or here. Since the Hamel dimension of all real sequences have the same cardinality $\mathfrak{c}$ and assuming the Continuum hypothesis, we can find a linear bijection $\phi$ from $l^{2}$ to the space of all sequences, say $\mathbb{R}^{\mathbb{N}}$. Hence, given a vector $w\in \mathbb{R}^{\mathbb{N}}$, define the norm $$\|w\|_{\mathbb{R}^{\mathbb{N}}} = \|\phi^{-1}(w)\|_{l^{2}}.$$ This norm define a complete Banach space. To define a complete Hilbert space it's just define a inner product in the same way.