The information about Basis (over infinite dimensional vector spaces) in the texts that i am reading assume that we are already dealing with a Hilbert Space and an orthonormal system. I am trying to understand how far we can abstract the notion of Schauder basis and what we can infer,
from: https://en.wikipedia.org/wiki/Schauder_basis
" Let V denote a Banach space over the field F. A Schauder basis is a sequence ${b_n}$ of elements of V such that for every element $v \in V$ there exists a unique sequence ${\alpha_n}$ of scalars in F so that
\begin{equation} {\displaystyle v=\sum _{n=0}^{\infty }\alpha _{n}b_{n}} \end{equation} where the convergence is understood with respect to the norm topology, i.e.,
\begin{equation} \lim _{n\to \infty }\left\|v-\sum _{k=0}^{n}\alpha _{k}b_{k}\right\|_{V}=0 \end{equation} Schauder bases can also be defined analogously in a general topological vector space. "
Suppose we have an infinite dimensional vector space $V$ such that every vector $v \in V$ can be expressed as the first equation. Here is my question:
- If we do not want to assume $V$ is a Banach space, what assumptions must be made on the vector space $V$ such that the set of coefficients ${\alpha_n}$ uniquely determine the vectors? i.e. if two vectors have the same set of coefficients, then the vectors are equal. is vector addition and scalar multiplication sufficient? i ask because convergence mentioned above is with respect to the norm topology, but the last line does not mention anything about the vector space having a norm.