i8n de la preuve

diff --git a/paper.tex b/paper.tex
index 4e407309a44605787351dec9043baa5b54742bc7..4a8bc4dbfa834bd3f6d674d8b7e44f662e08fc8b 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -757,13 +757,16 @@ $k$-th iterate of TSIRM.
 We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$.
 
 Each step of the TSIRM algorithm \\
+
+Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of vectors $S$. So,\\
 $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
 
 $\begin{array}{ll}
-& = \min_{x \in Vect\left(S_{k-s}, S_{k-s+1}, \hdots, S_{k-1} \right)} ||b-AS\alpha ||_2\\
-& = \min_{x \in Vect\left(x_{k-s}, x_{k-s}+1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
-& \leqslant \min_{x \in Vect\left( x_{k-1} \right)} ||b-Ax ||_2\\
-& \leqslant ||b-Ax_{k-1}||
+& = \min_{x \in span\left(S_{k-s}, S_{k-s+1}, \hdots, S_{k-1} \right)} ||b-AS\alpha ||_2\\
+& = \min_{x \in span\left(x_{k-s}, x_{k-s}+1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in span\left( x_{k-1} \right)} ||b-Ax ||_2\\
+& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k-1} ||_2\\
+& \leqslant ||b-Ax_{k-1}||_2 .
 \end{array}$
 \end{proof}
 
@@ -1069,4 +1072,3 @@ Curie and Juqueen respectively based in France and Germany.
 
 % that's all folks
 \end{document}
-