how to align equations with some text

Question

I want to align the equations with text. Though the equations are aligned as I want it to be but still the text is not aligned.

\begin{align}
 \mathscr{F}^\prime(x_0)^{-1}&\in L(Y,X) \quad &\textrm{for some } x_0\in{D}; \nonumber\\
\big\Vert{\mathscr{F}^\prime(x_0)^{-1}\mathscr{F}(x_0)}\big\Vert&\le {\color{red}\eta} &{}\nonumber
\\
\big\Vert{\mathscr{F}^\prime(x_0)^{-1}\big[\mathscr{F}^\prime(x)-\mathscr{F}^\prime(x_0)\big]}
\big\Vert &\le L_0
\left\Vert{x-x_0}\right\Vert\quad &\textrm{for all } x\in{D}; \nonumber\\
\big\Vert{\mathscr{F}^\prime(x_0)^{-1}\big[\mathscr{F}^\prime(x)-\mathscr{F}^\prime(y)\big]}
\big\Vert &\le L
\left\Vert{x-y}\right\Vert\quad &\textrm{for all } x,y\in{D}; \nonumber\\
h_{AH} = \xoverline{L}_{\eta} &\le \dfrac{1}{2} &{}
\intertext{and}
\xoverline{U}(x_0,\lambda_0) &\subseteq D, &{}\nonumber
\end{align}

Another question: is there any way to write the preceding equations more aesthetically? Thank your response.

Answer 1

You should put the text in the fourth column, which is left aligned:

\documentclass{article}

\usepackage{amsmath,mathrsfs,xcolor,mathtools}

\let\xoverline\overline % supply your definition

\begin{document}
\begin{align}
\mathscr{F}'(x_0)^{-1} 
  &\in L(Y,X)
  && \text{for some } x_0\in{D}; 
  \nonumber\\
\lVert\mathscr{F}'(x_0)^{-1}\mathscr{F}(x_0)\rVert 
  &\le \textcolor{red}{\eta} 
  \nonumber\\
\lVert\mathscr{F}'(x_0)^{-1}[\mathscr{F}'(x)-\mathscr{F}'(x_0)]\rVert 
  &\le L_0\lVert x-x_0\rVert 
  && \text{for all } x\in{D}; 
  \nonumber\\
\lVert\mathscr{F}'(x_0)^{-1}[\mathscr{F}'(x)-\mathscr{F}'(y)]\rVert 
  &\le L\lVert x-y\rVert 
  &&\text{for all } x,y\in{D}; 
  \nonumber\\
h_{AH} = \xoverline{L}_{\eta} 
  &\le \dfrac{1}{2}
\shortintertext{and}
\xoverline{U}(x_0,\lambda_0) 
  &\subseteq D, 
  \nonumber
\end{align}
\end{document}

I've removed all the \big commands (which should be \bigl and \bigr for left and right delimiters, but don't seem necessary here). Instead I used \lVert and \rVert to state precisely the nature of the symbols, whether opening or closing delimiters. The shorthand ' is equivalent to ^{\prime}. Finally, \shortintertext from the mathtools package requires less vertical space than \intertext.

It's not clear what the number refers to, the whole alignment or just the central equation.