The most commonly used representations are the coordinate and the momentum space representations. They define the completeness relations $$ \begin{eqnarray*} \mathbf{1}&=& \int d\mathbf{r} \:\vert\mathbf{r} \rangle \langle \mathbf{r}\vert, \:\: \langle \mathbf{r}\vert \mathbf{r'} \rangle = \delta ( \mathbf{r}-\mathbf{r'}) \\ \mathbf{1} &=& \int d\mathbf{k} \:\vert \mathbf{k}\rangle \langle \mathbf{k}\vert, \:\: \langle\mathbf{k}\vert \mathbf{k'} \rangle = \delta ( \mathbf{k}-\mathbf{k'}) \end{eqnarray*} $$ Here the basis states in both \( \mathbf{r} \)- and \( \mathbf{k} \)-space are dirac-delta function normalized. From this it follows that the plane-wave states are given by, $$ \langle\mathbf{r}\vert\mathbf{k} \rangle =\left(\frac{1}{2\pi}\right)^{3/2}\exp\left(i\mathbf{k\cdot r} \right) $$ which is a transformation function defining the mapping from the abstract \( \vert\mathbf{k}\rangle \) to the abstract \( \vert\mathbf{r}\rangle \) space.