In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \operatorname{SL}_n(\mathbb{C})$) acting linearly on a finite dimensional $\mathbb{C}$ vector space. To be more precise, the result he proves is (essentially) the following:
Let $V$ be a finite dimensional $\mathbb{C}$ vector space, $v \in V \backslash \{0\}$ and $G = \operatorname{GL}_n(\mathbb{C})$ (or $G = \operatorname{SL}_n(\mathbb{C})$) seen as a Zariski-closed subgroup of $\operatorname{GL}(V))$. Then $0 \notin \overline{G.v}$ (closure in the Hermitian topology) if and only if forall one parameter subgroup $\varphi : \mathbb{C}^* \longrightarrow G$, we have $\mu(v,\varphi) \geq 0$ (where $\mu(v, \varphi)$ is the opposite of the smallest weight of the action of $\varphi$ on a non-zero component of $v$).
Let me first sketch the proof of Sury (which I find aesthetically appealing) and then ask my question. We proceed by double contraposition and proves that $0 \in \overline{G.v}$ if and only if there is a one parameter subgroup $\varphi : \mathbb{C}^* \longrightarrow G$ such that $\mu(v, \varphi) < 0$.
If there is a one parameter subgroup $\varphi : \mathbb{C}^* \longrightarrow G$ such that $\mu(v, \varphi) < 0$, then the various weights of the action of $\varphi$ on the components of $v$ are all strictly positive. As a consequence, $\lim_{t \rightarrow} \varphi(t).v = 0$ and finally $0 \in \overline{G.v}$.
The non trivial implication is the direct one. Assume that $0 \in \overline{G.v}$. I will deal with the case $G = \operatorname{GL}_n(\mathbb{C})$. The Cartan decomposition insures that $G = U(n) T_n U(n)$, where $U(n)$ is the group of unitary matrices and $T_n$ is the group of diagonal invertible matrices. Since $U(n)$ is a compact group, we have the equivalence: $$0 \in \overline{G.v} \iff \exists w \in G.v, \ 0 \in \overline{T_n.w}.$$ So, up to dealing with $w$ instead of $v$, one can assume that $0 \in \overline{T_n}.v$. The group $T_n$ being a torus which acts linearly on $V$, its action on $V$ can be diagonalized, so that there exists $(m_{i,j}) \in \mathbb{Z}^{s \times n}$ such that: $$\forall (t_1, \ldots, t_n) \in T_n, \ \operatorname{diag}(t_1, \ldots, t_n).v = \sum_{i=1}^{s} t_1^{m_{i,1}} \ldots t_n^{m_{i,n}} v_i,$$ where $v= \sum_{i=1}^{s} v_i$, $v_i \in V_i$ and $V = \bigoplus_{i=1}^{r}V_i$ is a decomposition of $V$ adapted to the action of $\varphi$. The hypothesis $0 \in \overline{T_n.v}$ easily (this is a small computation with sequences and limits) implies that there is no non-zero vector $(b_1, \ldots, b_s) \in \mathbb{R}^s$ with positive coordinates such that: $$\forall j \in \{1,\ldots, n\}, \ \sum_{i=1}^{s} b_i m_{i,j} = 0.$$ The Gordan Theorem then implies that there exists $(a_1, \ldots, a_n) \in \mathbb{R}^n$ such that: $$ \forall i \in \{1, \ldots, s\}, \sum_{j=1}^{n} a_j m_{i,j} > 0.$$ By density, one can choose $(a_1, \ldots, a_n) \in \mathbb{Q}^n$ and finally, up to scaling by a common denominator, $(a_1, \ldots, a_n) \in \mathbb{Z}^n$. If one now considers the one-parameter subgroup $\varphi : \mathbb{C}^* \longrightarrow G$ defined by: $$\forall t \in \mathbb{C}^*, \ \varphi(t).v = \operatorname{diag}(t^{a_1}, \ldots, t^{a_n}).v,$$ then one sees that $\mu(v, \varphi) < 0$, wich finishes the proof.
Question : is there a similar proof for the Hilbert-Mumford stability criterion? More precisely, I would like to know if there is a similarly elementary proof of the implication: $$ \textrm{forall one parameter subgroup $\varphi : \mathbb{C}^* \longrightarrow G, \ \mu(v, \varphi) > 0$} \implies \ \textrm{the orbit} \ G.v \ \textrm{is closed in $V$ for the Hermitian topology} $$
I have heard that some analysists use a version the Hilbert-Mumford stability criterion in infinite dimensional vector spaces. So I guess they don't prove it using the subtleties of proper algebraic maps as in the book Geometric Invariant Theory by Mumford, Fogarty and Kirwan. This lets me hope that there might be an elementary proof of the above-mentioned implication.
