Dirac notation when bra and ket aren't vectors
Wave function for a quantum system is a vector in a Hilbert space, which is a special kind of an inner product space. Specifically, for a spin-0 particle constrained to move in 1 spatial dimension the wave function $\psi(x)$ is an element of $L^2(\mathbf{R})$ space, i.e. the space of functions that are square-integrable on the whole real line $\mathbf{R} = (-\infty;+\infty)$. And the inner product between any two elements $\vec{\psi}_{i} = \psi_i(x)$ and $\vec{\psi}_{j} = \psi_j(x)$ of $L^2(\mathbf{R})$ space is indeed defined as follows,
$$
\newcommand{\inner}[2]{\langle{#1},{#2}\rangle}
\newcommand{\ket}[1]{\,\lvert{#1}\rangle}
\newcommand{\braket}[2]{\langle{#1}\vert{#2}\rangle}
\inner{\vec{\psi}_{i}}{\vec{\psi}_{j}} := \int\limits_{-\infty}^{+\infty} \psi_i^*(x) \psi_j(x)\, \mathrm{d} x \, .
$$



Just note here that, first, it is the matter of convention in which argument the inner product is linear: in physical literature it is often defined to be linear in the second argument, while in mathematically oriented literature it is usually defined to be linear in the first argument. Consequently, mathematicians usually define the inner product between any two elements $\psi_i$ and $\psi_j$ of $L^2(\mathbf{R})$ space in the following way,
$$
\inner{\vec{\psi}_{i}}{\vec{\psi}_{j}} := \int\limits_{-\infty}^{+\infty} \psi_i(x) \psi_j^*(x)\, \mathrm{d} x \, ,
$$
as it is in fact done in some Wikipedia articles referenced above.

And secondly, physicists usually prefer to use even a different notation, the so-called Dirac bra-ket notation, in which elements of a Hilbert space are denoted as $\ket{\psi_i}$ and $\ket{\psi_j}$ (rather than $\vec{\psi}_{i}$ and $\vec{\psi}_{j}$ as it is usually done in linear algebra), while their inner product is designated as $\braket{\psi_i}{\psi_j}$ (rather than $\inner{\vec{\psi}_{i}}{\vec{\psi}_{j}}$).