Given the concepts of orthogonality and unit vector in a normed inner space defined as:
Two vectors $x,y$ are orthogonal if $\langle x, y \rangle = 0$
A vector $x$ is a unit vector if $\|x\| = 1$.
Then, is the following definition of orthonormality
A set of vectors $\{v_1, v_2, \dots, v_n\}$ is orthonormal if $\langle v_i, v_j \rangle = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker Delta
equivalent to
A set of vectors $\{v_1, v_2, \dots, v_n\}$ is orthonormal if $\forall(v_i \ne v_j), v_i$ and $v_j$ are orthogonal and all $v_i$ are unit vectors
The Wikipedia page on orthonormality seems to state both these definitions, but they don't seem equivalent to me. Is the inner product of a unit vector with itself necessarily 1?