Paul Dirac had the idea of thinking of an inner product $\langle {\bf x} | {\bf y} \rangle$, written as a bracket, as a product of a "bra" $\langle {\bf x} |$ and a "ket" $| {\bf y} \rangle$. In the following videos, we go over this idea in four settings:
The first video shows how this works for ${\bf R}^n$ and ${\bf C}^n$ with standard innner products. Kets are column vectors $|{\bf y} \rangle = {\bf y} = \begin{pmatrix} y_1 \cr \vdots \cr y_n \end{pmatrix}$. Bras are rows, and $\langle {\bf x} | = (\bar x_1, \bar x_2, \ldots, \bar x_n) = \overline{ {\bf x}^T}$.
When we have a non-standard inner product, we introduce a metric matrix $G$ with $G_{ij} = \langle {\bf e}_i | {\bf e}_j \rangle$. This matrix carries all the information about what inner product we are using. Then $\langle {\bf x} | = \bar {\bf x}^T G$, not $\bar {\bf x}^T$. The operation "take the inner product of ${\bf x}$ with the input" depends both on ${\bf x}$ and on what our inner product is.
When we have an abstract vector space $V$, then we need a basis $\mathcal{B} = \{{\bf b}_1, {\bf b}_2, \ldots, {\bf b}_n \}$ to make $V$ look like ${\bf R}^n$ or ${\bf C}^n$. But typically the basis does not make the inner product on $V$ look like the standard inner product on ${\bf C}^n$. So we once again need a metric matrix. Defining $G_{ij} = \langle {\bf b}_i | {\bf b}_j \rangle$, we can represent kets (i.e. elements of $V$) as columns, and bras (i.e. elements of $V^*$) as rows: $$ | {\bf y} \rangle_{\mathcal B} = [{\bf y}]_{\mathcal{B}} = \begin{pmatrix} y_1 \cr \vdots \cr y_n \end{pmatrix},$$ where ${\bf y} = y_1 {\bf b}_1 + \cdots + y_n {\bf b}_n$. Meanwhile, $${}_{\mathcal B}\langle {\bf x} | = \overline{[{\bf x}]_{\mathcal B}^T} G,$$ and the inner product $\langle {\bf x} | {\bf y} \rangle$ can be computed as the product of the row ${}_{\mathcal B}\langle {\bf x} |$ and the column $|{\bf y} \rangle_{\mathcal B}$. In the final video, we work out this formalism and apply it to a space of functions.