# §1.1 Special Notation

(For other notation see Notation for the Special Functions.)

$x,y$ real variables. complex variable in §§1.2(i), 1.9–1.11, real variable in §§1.5–1.6. complex variable in §§1.9–1.11. integers. nonnegative integers, unless specified otherwise. inner, or scalar, product for real or complex vectors or functions. the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$. a testing function. action of distribution $\Lambda$ on test function $\phi$. degree. derivatives with respect to the variable, except where indicated otherwise. column vectors. the space of all $n$-dimensional vectors. or $[a_{i,j}]$ or $[a_{ij}]$ matrix with elements $a_{i,j}$ or $a_{ij}$. inverse of the square matrix $\mathbf{A}$ identity matrix determinant of the square matrix $\mathbf{A}$ trace of the square matrix $\mathbf{A}$ exponential of $\operatorname{tr}(\mathbf{A})$ adjoint of the square matrix $\mathbf{A}$ complex conjugate of the matrix $\mathbf{A}$ transpose of the matrix $\mathbf{A}$ Hermitian conjugate of the matrix $\mathbf{A}$ linear operator defined on a manifold $\mathcal{M}$ adjoint of $\mathcal{L}$ defined on the dual manifold ${\mathcal{M}}^{*}$

In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{*}$, $a$ being a complex number or a matrix; the Hermitian conjugate of $\mathbf{A}$ is usually being denoted $\mathbf{A}^{{\dagger}}$.