Notation

Notation	Description
\(f:\mathcal{X}\rightarrow \mathcal{Y}\)	The function \(f\) maps from space \(\mathcal{X}\) to the space \(\mathcal{Y}\)
\(\vvec{x}\in\mathbb{R}^d\)	\(\vvec{x}\) is a vector in the \(d\)-dimensional real valued vector space
\(\vvec{x}\in\mathbb{R}^d_+\)	\(\vvec{x}\in\mathbb{R}^d\) and \(\vvec{x}\geq \vvec{0}\)
\(D\in\mathbb{R}^{n\times d}\)	\(D\) is a real-valued matrix with \(n\) rows and \(d\) columns
\(D_{\cdot k}\)	Column with index \(k\) (the \(k\)th column) of matrix \(D\)
\(D_{i\cdot}\)	Row with index \(i\) (the \(i\)th row) of matrix \(D\)
\(\lVert \cdot \rVert\)	A norm (depending on the context it is either standing for any norm or for the default Euclidian/Frobenius/\(L_2\)- norm )
\(\lvert \cdot \rvert\)	The \(L_1\) norm (for matrices it denotes the element-wise \(L_1\) norm)
\(\lVert \cdot\rVert_p\)	The \(L_p\) norm (it’s only a proper norm for \(p\geq 1\), for matrices it denotes the element-wise \(L_p\) norm)
\(\mathrm{tr}(\cdot)\)	The trace function
\(\nabla f\)	The gradient of \(f\)
\(\frac{\partial}{\partial x_k}f(x)\)	The partial derivative of \(f\) subject to \(x_k\)
\(\mathbf{1}\)	The constant one vector (for example if \(\mathbf{1}\) is a two-dimensional vector, then \(\mathbf{1}=\begin{pmatrix}1\\1\end{pmatrix}\)
\(\langle \cdot, \cdot \rangle\)	Inner product, the vector inner product is \(\langle \vvec{x},\vvec{y}\rangle = \vvec{x}^\top \vvec{y}\) and the matrix inner product is \(\langle X,Y\rangle = \mathrm{tr}(X^\top Y)\)
\(\mathrm{inf}/\mathrm{sup}\)	The infimum or supremum of a sequence/function. The infimum is like the minimum, handling a special case where the minimum is not defined. So, if you don’t know what inf/sup is, then think of it as min/max.
\(\mathrm{diag}(x_1,\ldots,x_n)\)	The \(n\times n\) diagonal matrix having the vector \(\mathbf{x}\) on the diagonal.
\(\log\)	The natural logarithm (to the base of \(e\))
$\lvert \mathcal{X}\rvert	The cardinality of set \(\mathcal{X}\) (the number of elements in \(\mathcal{X}\))