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% Document Info
\author{LuciaZhang}
\pdfinfo{
  /Title (appendix-a-functional-analysis.pdf)
  /Creator (Cheatography)
  /Author (LuciaZhang)
  /Subject (Appendix A - Functional Analysis Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
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    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Appendix A - Functional Analysis Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{LuciaZhang} via \textcolor{DarkBackground}{\uline{cheatography.com/136048/cs/48024/}}}
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\noindent
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}LuciaZhang \\
  \uline{cheatography.com/luciazhang} \\
  \end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 28th April, 2026.\\
   Page {\thepage} of \pageref{LastPage}.
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{document}
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{1.84149 cm} x{3.13551 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Normed spaces and Banach spaces}}  \tn
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\SetRowColor{LightBackground}
Strong convergence & ||x\textasciitilde{}n\textasciitilde{} - x||\textasciitilde{}X\textasciitilde{} -\textgreater{} 0 \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Continuous embedding & X ⊂ Y and ||x||\textasciitilde{}Y\textasciitilde{} ≤ C||x||\textasciitilde{}X\textasciitilde{} \tn 
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% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Banach space = complete normed space} \tn 
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\mymulticolumn{2}{x{5.377cm}}{Closure = add all limits of convergent sequences from the set} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.24425 cm} x{3.73275 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Bounded linear operators}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{D(A) ⊂ X, N(A) ⊂ X, R(A) ⊂ Y} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{A ∈ L(X,Y) \textless{}=\textgreater{} A linear and bounded} \tn 
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% Row 2
\SetRowColor{LightBackground}
\seqsplit{Boundedness} & ||Ax||\textasciitilde{}Y\textasciitilde{} ≤ C||x||\textasciitilde{}X\textasciitilde{} \tn 
% Row Count 4 (+ 2)
% Row 3
\SetRowColor{white}
Operator norm & ||A|| = sup\textasciitilde{}x≠1\textasciitilde{} ||Ax||\textasciitilde{}Y\textasciitilde{}／||x||\textasciitilde{}X\textasciitilde{} \tn 
% Row Count 6 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Inverse bounded ⟺ c||x||\textasciitilde{}X\textasciitilde{} ≤ ||Ax||\textasciitilde{}Y\textasciitilde{} (c \textgreater{} 0)} \tn 
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\end{tabularx}
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\begin{tabularx}{5.377cm}{x{1.74195 cm} x{3.23505 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Convergence, Banach-Steinhaus}}  \tn
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\SetRowColor{LightBackground}
Pointwise & ∀x, A\textasciitilde{}n\textasciitilde{}x -\textgreater{} Ax ∈ Y \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Uniform & ||A\textasciitilde{}n\textasciitilde{} - A||\textasciitilde{}L(X,Y)\textasciitilde{} -\textgreater{} 0 \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
Banach-\{\{nl\}\}Steinhaus & pointwise bounded family =\textgreater{} sup\textasciitilde{}i\textasciitilde{} ||A\textasciitilde{}i\textasciitilde{}|| \textless{} ∞ \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.44333 cm} x{3.53367 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Hilbert spaces, weak convergence}}  \tn
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\mymulticolumn{2}{x{5.377cm}}{Hilbert space = Banach space + scalar product} \tn 
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Weak \seqsplit{convergence} & x\textasciitilde{}n\textasciitilde{}⇀x ⟺ \textless{}x\textasciitilde{}n\textasciitilde{},z\textgreater{} -\textgreater{}\{\{nl\}\} \textless{}x,z\textgreater{},∀z ∈ X \tn 
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\mymulticolumn{2}{x{5.377cm}}{Strong =\textgreater{} weak. In infinite dims: weak ⇏ strong} \tn 
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\mymulticolumn{2}{x{5.377cm}}{Weak + convergence of norms =\textgreater{} strong} \tn 
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\begin{tabularx}{5.377cm}{x{1.54287 cm} x{3.43413 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Riesz representation, adjoint operators}}  \tn
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\{\{width=40\}\} Riesz & λ(x) = \textless{}z\textasciitilde{}λ\textasciitilde{}, x\textgreater{} \tn 
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\SetRowColor{white}
Adjoint & \textless{}A\textasciicircum{}*\textasciicircum{}y, x\textgreater{}\textasciitilde{}X\textasciitilde{} = \textless{}y, Ax\textgreater{}\textasciitilde{}Y\textasciitilde{} \tn 
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Matrix case & A\textasciicircum{}*\textasciicircum{} = A\textasciicircum{}T\textasciicircum{} over ℝ \tn 
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Properties & (A\textasciicircum{}*\textasciicircum{})\textasciicircum{}*\textasciicircum{} = A, ||A\textasciicircum{}*\textasciicircum{}|| = ||A||, ||A\textasciicircum{}*\textasciicircum{}A|| = ||A||\textasciicircum{}2\textasciicircum{} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.44333 cm} x{3.53367 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Orthogonality and projections}}  \tn
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\mymulticolumn{2}{x{5.377cm}}{U\textasciicircum{}⊥\textasciicircum{} = \{x ∈ X : \textless{}x,u\textgreater{} = 0 ∀u ∈ U\}} \tn 
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\mymulticolumn{2}{x{5.377cm}}{U\textasciicircum{}⊥\textasciicircum{} is always closed} \tn 
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If U closed: & X = U ⊕ U\textasciicircum{}⊥\textasciicircum{} \tn 
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If U not closed: & (U\textasciicircum{}⊥\textasciicircum{})\textasciicircum{}⊥\textasciicircum{} = cl(U) \tn 
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\SetRowColor{LightBackground}
Projection & z = P\textasciitilde{}U\textasciitilde{}x \{\{nl\}\} ⟺ z ∈ U and x - z ∈ U\textasciicircum{}⊥\textasciicircum{} \tn 
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\begin{tabularx}{5.377cm}{x{2.23965 cm} x{2.73735 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Range/kernel identities}}  \tn
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\SetRowColor{LightBackground}
R(A)\textasciicircum{}⊥\textasciicircum{} = N(A\textasciicircum{}*\textasciicircum{}) & N(A\textasciicircum{}*\textasciicircum{})\textasciicircum{}⊥\textasciicircum{} = cl(R(A)) \tn 
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R(A\textasciicircum{}*\textasciicircum{})\textasciicircum{}⊥\textasciicircum{} = N(A) & N(A)\textasciicircum{}⊥\textasciicircum{} = cl(R(A\textasciicircum{}*\textasciicircum{})) \tn 
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\mymulticolumn{2}{x{5.377cm}}{Kernel closed, range may be non-closed} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
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\begin{tabularx}{5.377cm}{x{1.4931 cm} x{3.4839 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Orthonormal systems and bases}}  \tn
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\SetRowColor{LightBackground}
Orthonormal & \textless{}u\textasciitilde{}i\textasciitilde{},u\textasciitilde{}j\textasciitilde{}\textgreater{} = δ\textasciitilde{}ij\textasciitilde{} \tn 
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\SetRowColor{white}
Bessel & ∑\textasciitilde{}j\textasciitilde{}|\textless{}x,u\textasciitilde{}j\textasciitilde{}\textgreater{}|\textasciicircum{}2\textasciicircum{} \textless{}= ||x||\textasciicircum{}2\textasciicircum{} \tn 
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\SetRowColor{LightBackground}
ONB expansion & x = ∑\textasciitilde{}j\textasciitilde{}\textless{}x,u\textasciitilde{}j\textasciitilde{}\textgreater{}u\textasciitilde{}j\textasciitilde{} \tn 
% Row Count 5 (+ 2)
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\SetRowColor{white}
Projection & P\textasciitilde{}U\textasciitilde{}x = ∑\textasciitilde{}j\textasciitilde{}\textless{}x,u\textasciitilde{}j\textasciitilde{}\textgreater{}u\textasciitilde{}j\textasciitilde{} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\begin{tabularx}{5.377cm}{x{0.9954 cm} x{3.9816 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Compact operators}}  \tn
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Compact & bounded sequence x\textasciitilde{}n\textasciitilde{} -\textgreater{} Ax\textasciitilde{}n\textasciitilde{} has a convergent subsequence \tn 
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\SetRowColor{white}
\seqsplit{Equivalent} & weak convergence in X -\textgreater{} strong convergence after A in Y \tn 
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\mymulticolumn{2}{x{5.377cm}}{Finite-dimensional range =\textgreater{} compact} \tn 
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\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{I\textasciitilde{}d\textasciitilde{} compact ⟺ dim(X) \textless{} ∞} \tn 
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\mymulticolumn{2}{x{5.377cm}}{A,S ∈ L(X,Y), A or S compact =\textgreater{} A∘S compact} \tn 
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\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{A compact ⟺ A\textasciicircum{}*\textasciicircum{} compact} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{0.4977 cm} p{0.4977 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Example: integral operators}}  \tn
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\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Kx(t) =  ∫\textasciitilde{}0\textasciitilde{}\textasciicircum{}1\textasciicircum{} k(s,t)x(s)ds} \tn 
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% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{Approximate k by piecewise-constant k\textasciitilde{}n\textasciitilde{}} \tn 
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\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{K\textasciitilde{}n\textasciitilde{} finite rank =\textgreater{} K\textasciitilde{}n\textasciitilde{} compact; \{\{nl\}\} K\textasciitilde{}n\textasciitilde{} -\textgreater{} K in operator norm =\textgreater{} K compact.} \tn 
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\mymulticolumn{2}{x{5.377cm}}{Self-adjoint ⟺ k(s,t) = k(t,s) a.e. (real case)} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
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\begin{tabularx}{5.377cm}{x{2.18988 cm} x{2.78712 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Spectral theorem (for compact, K=K\textasciicircum{}*\textasciicircum{})}}  \tn
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\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Kx = ∑\textasciitilde{}n\textasciitilde{}λ\textasciitilde{}n\textasciitilde{}\textless{}x,u\textasciitilde{}n\textasciitilde{}\textgreater{}u\textasciitilde{}n\textasciitilde{}, with λ\textasciitilde{}n\textasciitilde{} -\textgreater{} 0} \tn 
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% Row 1
\SetRowColor{white}
Ku\textasciitilde{}n\textasciitilde{} = λ\textasciitilde{}n\textasciitilde{}u\textasciitilde{}n\textasciitilde{}, & \{u\textasciitilde{}n\textasciitilde{}\} ONB for cl(R(K)) \tn 
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% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{||K|| = |λ\textasciitilde{}1\textasciitilde{}| when ordered by |λ\textasciitilde{}1\textasciitilde{}|≥|λ\textasciitilde{}2\textasciitilde{}|≥⋯≥0} \tn 
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% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{Inverse danger: divide by small λ\textasciitilde{}n\textasciitilde{}} \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}