What do the following symbols mean in this paper? Due to my limited knowledge, when I read the paper by Mattia Zorzi. I am confused about its notation. Hope you can help me explain.
See Zorzi, Mattia; Ticozzi, Francesco; Ferrante, Augusto, Minimum relative entropy for quantum estimation: feasibility and general solution, IEEE Trans. Inf. Theory 60, No. 1, 357-367 (2014). ZBL1364.81074.,

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*In equation 29, the authors wrote $\delta \mathcal{L}(\rho, \lambda; \delta \rho)$, I don’t know the meaning of the semicolon in brackets, i.e, $; \delta \rho$.

*In Equation 36, the authors wrote $\left\langle\lambda^{\perp}, \bar{f}\right\rangle$, where $\lambda=\left[\begin{array}{lll}\lambda_{1} & \ldots & \lambda_{m}\end{array}\right]^{T} \in \mathbb{R}^{m}$ and $\bar{f}=\left[\begin{array}{lll}\bar{f}_{1} & \ldots & \bar{f}_{m}\end{array}\right]^{T}$. I don’t know what the $\lambda^{\perp}$ means.

*In mathematics, what do $\operatorname{Range}$ and $\operatorname{range}$ stand for respectively. For example, the authors wrote  "note that $\left\langle\lambda^{\perp}, \bar{f}\right\rangle=0$ for each $\lambda^{\perp} \in[\text { Range } L]^{\perp}$" and "Notice that in order to have bounded values of the entropy it is only necessary that range $(\rho) \subseteq \operatorname{range}(\tau)$".

Thanks for your help!
 A: First thing to say, your question is slightly not about calculus of variations, because authors deal with finite-dimensional optimizations (minimizing functions with finite-dimensional argument), whereas calculus of variations is about infinite-dimensional arguments (for example, functions).

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*As it said below that equation, $\delta \rho$ means “direction” in $\mathcal{H}_n$, which is the space of all Hermitian matrices of dimension $n$. When you think about derivative in ordinary multidimensional calculus (with vectors as function arguments) you consider the change of some function $f(\vec{x})$ in some direction $\vec{h}$, that is $f(\vec{x}+\vec{h})$, and in given problem you minimize $\mathcal{L}(\rho, \lambda)$ over matrices, so direction also has to be a matrix.

*I think that $\lambda^\perp$ is just an authors’ way to note that $\lambda$ comes from the set $[\mathrm{Range} \;L]^\perp$, which is an “orthogonal complement” of $\mathrm{Range} \; L$ – in general, set of all vectors orthogonal to each vector of set $\mathrm{Range} \; L$, and $L$ is defined as linear operator from $\mathcal{H}_n$ to $\mathbb{R}^n$.

*For any function, range is the set of all values it can take, and linear operator (matrix) $A$ is a special case of function: $\mathbb{R}^n \ni x \mapsto Ax \in \mathbb{R}^m$. For Hermitian matrix $\mathcal{H}: \mathbb{R}^n \to \mathbb{R}^n$, eigenvectors form a basis of $\mathbb{R}^n$, so each $v \in \mathbb{R}^n$ is uniquely represented as $v = v_1 + v_2$, where $v_1 \in \mathrm{Range} \; \mathcal{H}$ and $v_2 \in \mathrm{Ker} \; \mathcal{H} = \left\{x \in \mathbb{R}^n : \mathcal{H}x = 0 \right\}$. Let $u$ be a vector such that $\mathcal{H} u=0$, and $v$ be any vector. Thus,
$\langle u, \mathcal{H} v\rangle=\left\langle \mathcal{H}^{*} u, v\right\rangle=\langle \mathcal{H} u, v\rangle=\langle 0, v\rangle=0$. As eigenvectors of Hermitian matrix corresponding to different eigenvalues are orthogonal, it follows that
$\mathrm{Range} \; \mathcal{H}$ is spanned by eigenvectors with non-zero eigenvalues and $\mathrm{Ker} \; \mathcal{H}$ - with zero eigenvalues.

EDIT: added some detailed information to 3.
