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Does "shift of terms" in a Faber–Schauder series expansion of f ∈ C[0,1] produce an element of C[0,1]?

# Question
Let $\{e_n : n=0, 1, \dots \}$ be a Faber–Schauder basis (See Megginson, 4.1.11 Example). Is the following function from $C[0,1] \times \mathbb{R}$ to $C[0,1]$ well-defined:

$$

(f, \beta) = (\sum \alpha_n e_n, \beta) \mapsto \beta e_0 + \sum \alpha_n e_{n+1}  

$$

The problem is in the shift in $\sum \alpha_n e_{n+1}$. I don't see how we can guarantee that $\sum \alpha_n e_{n+1}$ represents some function from $C[0,1]$.


# Context
I was wondering about alternative ways to set up a linear homeomorphism between $C[0,1] \times \mathbb{R}$ and $C[0,1]$. A classical way is to define the space $E$ with $C[0,1] = E \times c$. From this the required isomorphism follows. This is due to Banach (see: Theory of linear operations, Section Products of Banach spaces).

In terms of “the structure of space of scalars" I just ask whether such space is "right shift-invariant". With this wording, we can also ask a more general question:

What can we say about the structure of a sequence of scalars for a given Banach space and Shauder basis?

# Trivial observation
Let $c$ be a space of converging sequences with the supremum norm. Fix the Schauder basis  $\{ (0,\dots, 0,1 (n \text{-th place}),0, \dots) : n =1,2, \dots \}$. Then the corresponding space of sequences is shift invariant. That is
$$
\sum \alpha_n e_n \in c \implies \sum \alpha_n e_{n+1} \in c.
$$  

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I will be glad for thoughts, comments, and literature. Thanks!