Basis for a vector space and normed space

Question

While defining the basis for a vector space we impose two conditions (linearly independence and spaning) for the basis set. I am unable to see the condition of linearly independence in the case of schauder basis for a normed space. Can any body explain it to me why there is no need of being linearly independence for basis set in case of normed spaces and what is the role of linearly independence in case of vector space?

Answer 1

Schauder bases are linearly independent. Indeed, let $(e_n)_{n=1}^\infty$ be a Schauder basis for a Banach space. If it were linearly dependent, then the zero vector would have two expansions contradicting uniqueness:

$$0= \sum_{k=1}^\infty 0\cdot e_k = \sum_{k=1}^\infty a_k e_k$$

where $a_k$ are eventually zero, but some of them are non-zero and add up to 0 when multiplied by $e_k$.

However, linear independence in not really the primary issue when dealing with Schauder bases.