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On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$).

I am not sure if what I did was correct, but I started from using fact (and sketched proof of it) that in metric spaces separability is equivalent to existence of countable (topological) base.

Then, I was trying to show that if normed space have countable Hamel base $\mathscr{B}$, then it have countable open (topological) base. I wanted to do something like this:

Let's consider arbitrary $ x \in Lin \{ \mathscr{B} \} \cap {???}$ (there is more than countable many of them in $Lin \{ \mathscr{B} \}$... but maybe I can somehow fix it, for example taking only linear combinations for rational coefficients, or intersect it with some countable $???$ set?) and arbitrary positive rational $r \in \mathbb{Q}$. Then, we have (we have not (!!), but we will have after fixing what I mentioned above) countable number of balls $$B(x,r) = \{ y \in Lin \{ \mathscr{B} \} \cap {???} : ||x-y || < r \}$$ and all these balls makes topological basis of our vector space $X$.

I know that this sketch above looks a little chaotic, but can we accomplish something like that? (build countable number of balls, which make topological base of our space, using countable Hamel base in normed space?) If yes, how should I correctly construct them? Or there's also some other and easier way to do this task?


And here's where my second question, even more important (stronger) arrives. We can even say that this is the main purpose of this almost "essay" I just wrote above. After the exam I realized that I know nothing about connections between topological and algebraic bases in linear topological spaces. I also don't recall to hear anything about it on my course. I was trying to check in the internet (uncle Google), but cannot find anything useful (or I just lack awareness about it usefulness).

Do you know any dependences between algebraic and topological bases of the same space?

Do they always must to have equal powers?

Are there some interesting properties, connections, or possible differences in some of these properties between them?

How much does it differ between linear topological spaces and linear topological spaces which are metrizable, normable, etc.?

I would be glad to hear about any of things mentioned above or to receive some references, like some chapters in books, links, publications or other materials where I can find this matter discussed or explained.

Or maybe I am just exaggerating and there's not really much to tell about this all? There was nothing said about this on my course nor I cannot find this in my books or in internet, so I would be very glad to find something, which can give me full picture about this matter.

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I would want to better understand and find out something regarding second part of my question (about topological and algebraic bases) (in generality, not focusing only on Banach spaces), so I add 50 rep bounty. I know it is not much and I don't have too much, but I hope it will encourage someone who can know about that or can point to where I can find some constructive results on this matter ^^.

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Maybe you will find this interesting: people.math.gatech.edu/~heil/papers/bases.pdf. Also, a Hamel Basis is simply a basis in the sense of Linear algebra, and no Banach space can have a countable basis. Maybe you meant Schauder basis, in which case you have this solution? –  Luiz Cordeiro Jun 24 at 18:36
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Thank you, I will look into this. In this task I meant normed space (which don't need to be Banach, thus don't need to have uncountable basis) with countable Hamel basis, but thank you very much for the references. –  Kusavil Jun 24 at 18:50
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I think that if you take a basis $B$ such that for each $b\in B$ you have $\|b\|=1$ and then take all rational linear combinations, you should be able to show that this set is dense in $X$. I.e., for a given $x=c_1b_1+\dots+c_nb_n$ and given $\varepsilon>0$ you should find $d_1,\dots,d_n$ such that $x$ is $\varepsilon$-closed to $y=d_1b_1+\dots+d_nb_n$. –  Martin Sleziak Jun 24 at 19:17
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@Kusavil I guess I'm too used to reading Banach. Anyway, in the solution I referenced for Schauder bases, the solutions given also work for your case (and are in fact the same as Martin suggested). –  Luiz Cordeiro Jun 24 at 19:56
    
Thank you @ Luiz Cordeiro, @ Martin Sleziak :-) I thought it can be something like that but I had problem with formalizing it. I will try to write it down in formal way for myself tomorrow. –  Kusavil Jun 24 at 20:19

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