Inner product

We start with standard inner product, i.e.,

\begin{align} \left[ \begin{array}{ccc} a_1& a_2 &a_3 \end{array} \right] \cdot \left[ \begin{array}{c} b_1\\b_2\\b_3 \end{array} \right]=a_1b_1+a_2b_2+a_3b_3. \end{align}

Using Einstein summation convention, we write

\begin{align} \left[ \begin{array}{ccc} a_1& a_2 &a_3 \end{array} \right] \cdot \left[ \begin{array}{c} b_1\\b_2\\b_3 \end{array} \right]=a_ib_i. \end{align}

The number of components would be assumed from the context; hence it is not clearly written. In addition, double indices implies summation; hence summation symbol is not written.
Let $\boldsymbol{e}_1,\boldsymbol{e}_2\cdots$ be basis. An arbitrary vector can be represented as linear combinations of the basis such as

\begin{align} \boldsymbol{a}=a_1\boldsymbol{e}_1+a_2\boldsymbol{e}_2+a_3\boldsymbol{e}_3=a_i\boldsymbol{e}_i \end{align}


\begin{align} \boldsymbol{b}=b_1\boldsymbol{e}_1+b_2\boldsymbol{e}_2+b_3\boldsymbol{e}_3=b_i\boldsymbol{e}_i. \end{align}

When inner products between basis is given as

\begin{align} \boldsymbol{e}_i\cdot\boldsymbol{e}_j=\delta_{ij}, \end{align}

this implies that the basis are chosen as orthonormal.
Based on the inner-products between basis, standard inner product can be derived as follows:

\begin{align} \boldsymbol{a}\cdot\boldsymbol{b}=\left(a_i\boldsymbol{e}_i\right)\cdot\left(b_j\boldsymbol{e}_j\right)=a_ib_j\boldsymbol{e}_i\cdot\boldsymbol{e}_j=a_ib_j\delta_{ij}=a_ib_i. \end{align}

The standard inner product is referred to as Euclidean inner product.
Incidentally, when Eq. (5) is given, we can derive Eq. (1) using only properties of multiplications. Which means that, Eq. (5) and Eq. (1) correspond uniquely.
Generalization of standard inner product can be derived as follows. The basic idea is the when Eq. (5) is generalized, we can generalize the standard inner product.
When basis $\boldsymbol{g}_1\cdots$ are such like

\begin{align} \boldsymbol{g}_i\cdot\boldsymbol{g}_j\neq \delta_{ij}, \end{align}

an arbitrary vector is represented as

\begin{align} \boldsymbol{a}=a^1\boldsymbol{g}_1+a^2\boldsymbol{g}_2+a^3\boldsymbol{g}_3=a^i\boldsymbol{g}_i, \end{align}
\begin{align} \boldsymbol{b}=b^1\boldsymbol{g}_1+b^2\boldsymbol{g}_2+b^3\boldsymbol{g}_3=b^i\boldsymbol{g}_i. \end{align}

The reason why the indices are superscripts is explained in Introduction to Surface Theory.
When inner products between such general basis are given as

\begin{align} \boldsymbol{g}_i\cdot\boldsymbol{g}_j=g_{ij}, \end{align}

generalized inner product can be derived by using only properties of multiplications as:

\begin{align} \left<\boldsymbol{a},\boldsymbol{b}\right>=\left(a^i\boldsymbol{g}_i\right)\cdot\left(b^i\boldsymbol{g}_i\right)=a^ib^j\boldsymbol{g}_i\cdot\boldsymbol{g}_j=a^ib^jg_{ij}. \end{align}

It can be known that standard inner product is a special case of generalized inner product, i.e., $g_{ij}=\delta_{ij}$.