
\newlength{\fixboxwidth}
\setlength{\fixboxwidth}{\marginparwidth}
\addtolength{\fixboxwidth}{+26.8pt}
\newcommand{\fix}[1]{\marginpar{%
    \fbox{\parbox{\fixboxwidth}{\raggedleft\tiny #1}}}}



%\newcommand{\remark}[1]{}
\newcommand{\eins}{\boldsymbol{1}}

\newcommand{\T}{\mathbb{T}}    % Sonderzeichen f�r

\newcommand{\C}{\mathbb{C}}    % Sonderzeichen f�r

\newcommand{\N}{\mathbb{N}}    % Sonderzeichen f�r
\newcommand{\R}{\mathbb{R}}    % Sonderzeichen f�r
\newcommand{\Rp}{\mathbb{R}^+}    % Sonderzeichen f�r
\newcommand{\Rn}{\mathbb{R}^d}
\newcommand{\oR}{\overline\R}    %   % Sonderzeichen f�r
\newcommand{\K}{\mathbb{K}}    % Sonderzeichen f�r
\newcommand{\Z}{\mathbb{Z}}    % Sonderzeichen f�r
\def \P {\mathbb{P}}    % Sonderzeichen f�r
\newcommand{\E}{\mathbb{E}}    % Sonderzeichen f�r


\newcommand{\opa}{\mbox{\rm Pr}^*}



\newcommand{\So}[1]{{\sf SO(\mbox{$#1$})}}
\newcommand{\On}[1]{{\sf ON(\mbox{$#1$})}}
\newcommand{\Gl}[1]{{\sf GL(\mbox{$#1$})}}

\newcommand{\ca}[1]{\mbox{${\mathcal #1}$}}
%\newcommand{\rem}[1]{}

\newcommand{\iso}{=\hspace{-0.8em}^{^1}\hspace{0.4em}}
\newcommand{\isoin}{\hookrightarrow\hspace{-1em}^{^1}\hspace{0.6em}}




%\newcommand{\BlackBox}{\rule{1.5ex}{1.5ex}}  % end of proof
\newenvironment{pf}{\par\noindent{\it Proof:\ }}{\hfill\BlackBox\vspace*{2mm}}




\def \B         {{\cal B}}                % Borelsche s-Algebra

\def \Om        { \Omega }
\def \om        { \omega }
\def \lb        { \lambda }
\def \a         { \alpha }
\def \b         { \beta }
\def \g         { \gamma }
\def \e         { \varepsilon }
\def \r         { \rho }
\def \s         { \sigma }
\def \vs        { \varsigma }
\def \t         { \tau }
\def \et        { \tilde{\eta}}
\def \d         { \delta }
\def \D         { \Delta }
\def \p         { \varphi }
\def \x         { \xi }
\def \Ph         { \Phi }


\newcommand{\ld}{\log_2}
\newcommand{\loq}{\log_m}
\newcommand{\ed}{\exp_2}

\newcommand{\edot}{\!\cdot\!}

\newcommand{\mez}{{2^{m-1}-2}}
\newcommand{\mee}{{2^{m-1}-1}}
\newcommand{\me}{{2^{m-1}}}
\newcommand{\iin}{i=0,\dots,\mee}
\newcommand{\jin}{j\in\{-1,1\}}

\newcommand{\ob}{\overline b_{\e,\d}}
\newcommand{\ub}{\underline b_{\e,\d}}
\newcommand{\xed}[1]{X_{#1,\d}}
\newcommand{\xpedn}{X_{\e,\d,n}^+}
\newcommand{\xmedn}{X_{\e,\d,n}^-}
\newcommand{\xnedn}{X_{\e,\d,n}^0}



\newcommand{\Sim}{\ \sim \ }
\newcommand{\Preceq}{\ \preceq \ }
\newcommand{\Succeq}{\ \succeq \ }
\newcommand{\Leq}{\ \leq \ }
\newcommand{\Geq}{\ \geq \ }
\newcommand{\Eq}{\ = \ }
\newcommand{\Defi}{\ := \ }


\newcommand{\Lfloor}{\left\lfloor}
\newcommand{\Rfloor}{\right\rfloor}









\newcommand{\Int}{\int\limits}
\newcommand{\Sum}{\sum\limits}

\DeclareMathOperator{\co}{co}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\diam}{diam}
\DeclareMathOperator{\aco}{aco}

\newcommand{\modc}[2]{\mbox{$\omega (#1,#2)$}}


\newcommand{\im}[1]{\mbox{${\rm {im}} \ #1$}}
\newcommand{\Ker}[1]{\mbox{${\mathfrak {Ker}} \ #1$}}
\newcommand{\rank}[1]{{\rm rank} \ #1}
%\newcommand{\rank}[1]{\mbox{\rm rank $#1$}}
\newcommand{\codim}[1]{\mbox{\rm codim $#1$}}
\newcommand{\vol}{\mbox{\rm vol}}
\DeclareMathOperator{\spann}{span}
\newcommand{\card}{\mbox{\rm card }}
\newcommand{\dn}[2]{\mbox{$d_{\| .\|}(#1,#2)$}}

\newcommand{\ol}[1]{\overline{#1}}

\DeclareMathOperator{\vcdim}{VC-dim}
\DeclareMathOperator{\pot}{pot}


\newcommand{\id}{\mbox{${\rm {id}}$}}

\newcommand{\posku}[2]{\mbox{$\ca W_{#1}(#2)$}}
\newcommand{\einku}[2]{\mbox{$\ca W_{#1}^1(#2)$}}

\newcommand{\btens}{\bigotimes}
\newcommand{\tens}{\otimes}
\newcommand{\xtens}[1]{\otimes_{_{\triangle_{#1}}}}
\newcommand{\afalt}{\circledast}


%Normen und R�ume

\newcommand{\lt}[1]{\mbox{${\ell}_2 (#1)$}}

\newcommand{\lto}{\mbox{${\ell}_2$}}
%Der Raum Lo

\newcommand{\Lo}{\mbox{${L}_o (\mu)$}}
\newcommand{\Lom}[1]{\mbox{${L}_o (#1)$}}
\newcommand{\Lox}[1]{\mbox{${L}_o (\mu,#1)$}}
\newcommand{\Loxinf}[1]{{\sf L\mbox{$_o (\mu,#1)$}}}
\newcommand{\Lof}{\mbox{${L}_o (\mu,F)$}}
\newcommand{\onorm}[1]{\mbox{$\left\Vert #1 \right\Vert_{_0}$}}

%Der Raum Lps

\newcommand{\Lps}{\mbox{${L}_{p,\infty} (\mu)$}}
\newcommand{\Lqs}{\mbox{${L}_{q,\infty} (\mu)$}}
\newcommand{\Lpsm}[1]{\mbox{${L}_{p,\infty} (#1)$}}
\newcommand{\Lxsm}[2]{\mbox{${L}_{#1,\infty} (#2)$}}
\newcommand{\psno}[1]{\mbox{$\left\Vert #1 \right\Vert_{p,\infty}$}}
\newcommand{\qsno}[1]{\mbox{$\left\Vert #1 \right\Vert_{q,\infty}$}}


% Der Raum Lp

\newcommand{\Lp}{\mbox{${L}_{p} (\mu)$}}
\newcommand{\Lpm}[1]{\mbox{${L}_{p} (#1)$}}
\newcommand{\Lx}[1]{\mbox{${L}_{#1} (\mu)$}}
\newcommand{\Lxn}[1]{\mbox{${L}_{#1} (\nu)$}}
\newcommand{\Lpx}[3]{\mbox{${L}_{#1} (#2,#3)$}}
\newcommand{\Le}{\mbox{${L}_{1} (\mu)$}}
\newcommand{\Li}{\mbox{${L}_{\infty} (\mu)$}}
\newcommand{\Lpmx}[2]{\mbox{${L}_{#1} (#2)$}}
\newcommand{\Lpmxge}[2]{\mbox{${L}_{#1}^{^{\geq 0}} (#2)$}}
\newcommand{\Lpmxpo}[2]{\mbox{${L}_{#1}^{^{> 0}} (#2)$}}
\newcommand{\wLpx}[3]{\mbox{$w^*$-$L_{#1} (#2,#3')$}}
\newcommand{\wsLpx}[3]{\mbox{$w^*$-$\ca L_{#1} (#2,#3')$}}
\newcommand{\wN}[3]{\mbox{$w^*$-$\ca N_{#1} (#2,#3')$}}


\newcommand{\pnorm}[1]{\mbox{$\left\Vert #1 \right\Vert_{p}$}}
\newcommand{\norm}[1] {\Bigl\Vert #1 \Bigr\Vert}
\newcommand{\Norm}[1] {\left\Vert #1 \right\Vert}
\newcommand{\snorm}[1] {\Vert #1 \Vert}

\newcommand{\enorm}[1]{\mbox{$\left\Vert #1 \right\Vert_{E}$}}
\newcommand{\fnorm}[1]{\mbox{$\left\Vert #1 \right\Vert_{F}$}}
\newcommand{\xnorm}[3]{\mbox{$\left\Vert #1 \right\Vert_{{#2}}^{#3}$}}
\newcommand{\ynorm}[2]{\mbox{$\left\Vert #1 \right\Vert_{{#2}}$}}
\newcommand{\betr}[1]{\mbox{$\left\vert #1 \right\vert$}}

\newcommand{\M}[4]{\mbox{$\ca M (\Lpx #1 G #3,\Lpx #2 G #4)$}}

\newcommand{\einorm}[1]{\mbox{$\left\Vert #1 \right\Vert_{1}$}}
\newcommand{\inorm}[1]{\mbox{$\left\Vert #1 \right\Vert_{\infty}$}}
\newcommand{\isnorm}[1]{\mbox{$\bigl\Vert #1 \bigr\Vert_{\infty}$}}

\newcommand{\Coo}[1]{\mbox{${C}_{oo} (#1)$ }}
\newcommand{\supp}[1]{\mbox{{\rm supp} $#1$}}
\newcommand{\sL}[2]{\mbox{${\cal L}(#1,#2)$ }}
\newcommand{\sLp}[1]{\mbox{${\cal L}_p(\mu)$ }}


