Let $I$ be any non-empty set, one can consider the set $2^I_{fin} := \{ J \in 2^I \; : \; J \text{ is finite } \}$ So basically $2^I_{fin}$ consists of all the finite subset of $I$. Then the ordered set $(2^J_{fin} , \subseteq )$ is directed. Moreover if $\mathbb{V}$ is a topological vector space and $(v_i)_{i \in I}$ is a sequence in $\mathbb{V}$ with index set $I$, then it is well defined (although it may not exist) the sum $\sum_{i \in I}{v_i} := \lim_{J \in 2^{I}_{fin} }{\sum_{j \in J}{v_j} }$
So basically I'm taking the limit of the Net
$S_J :=\sum_{j \in J}{v_j}$
where $J \in 2^I_{fin}$.
Having this in mind let's define a notion of a basis in a topological vector space.
Let $\mathbb{V}$ be a topological vector space over the field $\mathbb{K}$, let $S \subset \mathbb{V}$, then
1)$S$ is said to be weakly linearly independent if and only if for any finite subset $S'$ of $S$ $S'$ is linearly independent.
- $S$ is said to be strongly linearly independent if and only if the only sequence $(c_s)_{s \in S}$ of scalars in $\mathbb{K}$ such that
$\sum_{s \in S}{c_s s} = \underline{0}$
Is the null sequence $c_s = 0 \; \forall s \in S$
- $S$ is said to be almost strongly linearly independent if and only if for any at most countable subset $S'$ of $S$ the only sequence $(c_s)_{s \in S'}$ of scalars in $\mathbb{K}$ such that
$\sum_{s \in S'}{c_s s} = \underline{0}$
Is the null sequence $c_s = 0 \; \forall s \in S'$
$S$ is said to be a weak system of generators if and only if $\overline{span(S)} = \mathbb{V}$ where $span(S)$ is the set of finite linear combination of vectors of $S$ (meaning that all but a finite number of the $c_s$ are equal to 0 ).
$S$ is said to be a strong system of generators if and only if for any $v \in \mathbb{V}$ there exists at least one sequence of scalars $(c_s)_{s \in S}$ such that
$v = \sum_{s \in S}{c_s s}$
- $S$ is said to be a perfect system of generators if and only if for any $v \in \mathbb{V}$ there exists at least one finite subset $S'$ of $S$ and one finite sequence $(c_s)_{s \in S'}$ such that
$v = \sum_{s \in S'}{c_s s}$
- $S$ is said to be an almost perfect system of generators if and only if for any $v \in \mathbb{V}$ there exists at least one at most countable subset $S'$ of $S$ and one sequence $(c_s)_{s \in S'}$ such that
$v = \sum_{s \in S'}{c_s s'}$
Clearly a basis need to be a system of generators and linearly independent. By chosing any definition of linearly independent and system of generators In particular because there are 3 definitions of linear independence and 4 definitions of system of generators there are in total 12 = 4 \times 3 possible definition of a basis. Obviuos results holds, for example let $SG$ stand for "system of generators" and $LI$ stand for linearly independent then one has
perfect $SG \, \implies $ almost perfect $SG \, \implies$ strong $SG \, \implies$ weak $SG$
And also
strongly $LI \, \implies $ almost strongly $LI \, \implies$ weakly $LI$
Furthermore when $\mathbb{V}$ has a finite dimension all the definitions are equivalent.
Certain choiches of definitions gives the definitions of Schauder basis (you need to add the hypothesis that S is at most countable) and Hamel basis.
My question is :
Can you give me an example (or prove that it doesn't exists) of a set $S$ which is strongly LI and is a weak SG but is not a strong SG ?
For each possible definition of basis does the theorems of the uniqueness of the cardinality of the base and of the existance of the base always holds? If yes how can I prove them? If not can you give me a counterexample?
Does there exists a more than countable set $S$ which is strongly LI and a strong SI? If yes give me an example, if not prove that it doesn't exists.
Which type of basis does $L^\infty(\mathbb{R})$ admits? Which one it doesn't?