We characterize when the norm of the sum of elements 
in a Banach space is equal to the sum of their norms 
and give its applications to unital Banach algebras. 
Related to a recent result due to Barraa and Boumazgour, 
we give another equivalent condition of it in terms of 
linear functional on a unital $C^*$-algebra. Consequently, 
we have the following result. 
 \par
 For elements $a$ and $b$ in a unital $C^*$-algebra, 
the following statements are mutually equivalent:
 \par
 (i) $\|a+b\|=\|a\|+\|b\|.$
 \par
 (ii) There exists a norm one linear functional $f$ 
such that $f(a)=\|a\|$ and $f(b)=\|b\|.$
 \par
 (iii) There exists a state $f$ such that 
$f(a^*b)=\|a\|\|b\|.$