Dirac notation is a notation to do calculations in a Hilbert space without needing to worry about the space’s representation. It is basically the lingua franca of quantum mechanics.
+In Dirac notation there are kets \(\ket{V}\) from the Hilbert space \(\mathbb{H}\) and bras \(\bra{V}\) from a dual \(\mathbb{H}'\) of the former. Crucially, the bras and kets are from different Hilbert spaces and therefore cannot be added, but every bra has a corresponding ket and vice versa.
+Bras and kets can only be combined in two ways: the inner product \(\braket{V | W}\), which returns a scalar, and the outer product \(\ket{V} \bra{W}\), which returns a mapping \(\hat{L}\) from kets \(\ket{V}\) to other kets \(\ket{V'}\), i.e. a linear operator. Recall that the Hilbert inner product must satisfy:
+\[\begin{aligned} + \braket{V | W} = \braket{W | V}^* +\end{aligned}\]
+So far, nothing has been said about the actual representation of bras or kets. If we represent kets as \(N\)-dimensional columns vectors, the corresponding bras are given by the kets’ adjoints, i.e. their transpose conjugates:
+\[\begin{aligned} + \ket{V} = + \begin{bmatrix} + v_1 \\ \vdots \\ v_N + \end{bmatrix} + \quad \implies \quad + \bra{V} = + \begin{bmatrix} + v_1^* & \cdots & v_N^* + \end{bmatrix} +\end{aligned}\]
+The inner product \(\braket{V | W}\) is then just the familiar dot product \(V \cdot W\):
+\[\begin{gathered} + \braket{V | W} + = + \begin{bmatrix} + v_1^* & \cdots & v_N^* + \end{bmatrix} + \cdot + \begin{bmatrix} + w_1 \\ \vdots \\ w_N + \end{bmatrix} + = v_1^* w_1 + ... + v_N^* w_N +\end{gathered}\]
+Meanwhile, the outer product \(\ket{V} \bra{W}\) creates an \(N \cross N\) matrix:
+\[\begin{gathered} + \ket{V} \bra{W} + = + \begin{bmatrix} + v_1 \\ \vdots \\ v_N + \end{bmatrix} + \cdot + \begin{bmatrix} + w_1^* & \cdots & w_N^* + \end{bmatrix} + = + \begin{bmatrix} + v_1 w_1^* & \cdots & v_1 w_N^* \\ + \vdots & \ddots & \vdots \\ + v_N w_1^* & \cdots & v_N w_N^* + \end{bmatrix} +\end{gathered}\]
+If the kets are instead represented by functions \(f(x)\) of \(x \in [a, b]\), then the bras represent functionals \(F[u(x)]\) which take an unknown function \(u(x)\) as an argument and turn it into a scalar using integration:
+\[\begin{aligned} + \ket{f} = f(x) + \quad \implies \quad + \bra{f} + = F[u(x)] + = \int_a^b f^*(x) \: u(x) \dd{x} +\end{aligned}\]
+Consequently, the inner product is simply the following familiar integral:
+\[\begin{gathered} + \braket{f | g} + = F[g(x)] + = \int_a^b f^*(x) \: g(x) \dd{x} +\end{gathered}\]
+However, the outer product becomes something rather abstract:
+\[\begin{gathered} + \ket{f} \bra{g} + = f(x) \: G[u(x)] + = f(x) \int_a^b g^*(\xi) \: u(\xi) \dd{\xi} +\end{gathered}\]
+This result makes more sense if we surround it by a bra and a ket:
+\[\begin{aligned} + \bra{u} \!\Big(\!\ket{f} \bra{g}\!\Big)\! \ket{w} + &= U\big[f(x) \: G[w(x)]\big] + = U\Big[ f(x) \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big] + \\ + &= \int_a^b u^*(x) \: f(x) \: \Big(\int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) \dd{x} + \\ + &= \Big( \int_a^b u^*(x) \: f(x) \dd{x} \Big) \Big( \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) + \\ + &= \braket{u | f} \braket{g | w} +\end{aligned}\]
+