Basic sequence- what is so special about it? Let $(x_n)$ be a Schauder basis of a vector space $X$. This means that the $span(x_n)$ is dense in $X$, right?
Then wikipedia introduces the notion of a $\textbf{basic sequence} $ when $(x_n)$ is a Schauderbasis of the closure of its linear span.
I don't understand this. The closure of its linear span is just the vector space $X$, isn't it? So how is this concept different from just saying that $(x_n)$ is a Schauder basis of $X$
 A: It appears that you misunderstood the concept of Schauder basis. It requires much more than having dense linear span (and being linearly independent). Namely, for every element $x$ of the space we must have a (unique) series of the form $\sum c_j x_j$ converging to it. This is more than saying $x$ is the limit of a sequence of some linear combinations of $x_j$. 
When building a series, you have to decide what the coefficient of each basis vector is, and stick with it. When building a sequence of linear combinations, you can use different coefficients for the same vector, in each term of the sequence. 
Standard example: the monomials $1,x,x^2,x^3,\dots,$ have dense linear span in $C[0,1]$, because we can approximate any continuous function by polynomials (Weierstrass's theorem). But if the monomials were a Schauder basis for $C[0,1]$,  every continuous  function would be  the sum of a power series. This is patently false: the sum of any power series is real-analytic, in particular $C^\infty$-smooth. 
Summary: since a Schauder basis is a very special kind of a spanning set, a basic sequence is a very special kind of a sequence.
