We denote by $I$ a subset of $\mathbb { N }$ having at least two elements and by $u = \left( u _ { i } \right) _ { i \in I }$ a sequence of unit vectors of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. $C(u)$ denotes the coherence parameter $C ( u ) = \sup \left\{ \left| \left\langle u _ { i } \mid u _ { j } \right\rangle \right| , ( i , j ) \in I ^ { 2 } , i \neq j \right\}$. Show that if $C ( u ) = 0$, then the set $\left\{ u _ { i } , i \in I \right\}$ is finite and give an upper bound for its cardinality.
We denote by $I$ a subset of $\mathbb { N }$ having at least two elements and by $u = \left( u _ { i } \right) _ { i \in I }$ a sequence of unit vectors of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. $C(u)$ denotes the coherence parameter $C ( u ) = \sup \left\{ \left| \left\langle u _ { i } \mid u _ { j } \right\rangle \right| , ( i , j ) \in I ^ { 2 } , i \neq j \right\}$.
Show that if $C ( u ) = 0$, then the set $\left\{ u _ { i } , i \in I \right\}$ is finite and give an upper bound for its cardinality.