# Is orthonormality equivalent to orthogonality and normalization in a normed inner product space?

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?

For any inner product space with inner product $$(\cdot, \cdot)$$, we define the norm on that space by $$\|x\| = \sqrt{(x,x)}$$. For more info, see here. With the norm so defined, if $$\|x\|= 1$$, then $$(x,x) = \|x\|^2 =1$$, and so the definitions are equivalent.