Suppose $a \in \ell^p$ and $\|a \|_p = 1$. Does this then imply that $| a_n | = 1$, for all $n$?

Question

Let $a \in \ell^p$.

Suppose $\|a \|_p = 1$, i.e.,

$$ \left( \sum_{n=1}^{\infty} |a_n|^p \right)^{1/p} = 1. $$

Does this, in fact, imply that $|a_n| = 1,$ for all $n$? If not, what exactly does it imply about the individual elements of the sequence $a$?

Answer 1

No. Consider $p=1$, and take $\sum_{n=1}^{\infty} \frac{1}{2^n}=1$.

If $a_i=c$ for any $x \in \mathbb R \setminus \{0\}$, then the series diverges.