In a Linear Vector space I can have an infinite number of vectors, given that I have an orthonormal basis set. If the LVS has infinite dimensionality then, I learnt that it needs to be square summable. So if I take a vector $V(x_{1}, x_{2},...)$ in this LVS and calculate the inner product as:
$Σ|x_{i}|^2 < ∞$
Does this hold true for ANY vector in the LVS? I could possibly give example that this sequence is convergent for certain vectors whilst divergent for others. Will it still be called an $ℓ^{2}$?