%% %% This is file `samplecards.tex', %% generated with the docstrip utility. %% %% The original source files were: %% %% flashcards.dtx (with options: `sample') %% %% FlashCards LaTeX2e Class for Typesetting Double Sided Cards %% Copyright (C) 2000 Alexander M. Budge %% %% This program is free software; you can redistribute it and/or modify %% it under the terms of the GNU General Public License as published by %% the Free Software Foundation; either version 2 of the License, or %% (at your option) any later version. %% %% This program is distributed in the hope that it will be useful, %% but WITHOUT ANY WARRANTY; without even the implied warranty of %% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the %% GNU General Public License for more details. %% %% You should have received a copy of the GNU General Public License %% along with this program (the file COPYING); if not, write to the %% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. %% %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! Double quote \" Hash (number) \# %% Dollar \$ Percent \% Ampersand \& %% Acute accent \' Left paren \( Right paren \) %% Asterisk \* Plus \+ Comma \, %% Minus \- Point \. Solidus \/ %% Colon \: Semicolon \; Less than \< %% Equals \= Greater than \> Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} %% \NeedsTeXFormat{LaTeX2e}[1996/12/01] \ProvidesFile{samplecards.tex} \documentclass[avery5388,grid,frame]{flashcards} \cardfrontstyle[\large\slshape]{headings} \cardbackstyle{empty} \begin{document} \cardfrontfoot{Functional Analysis} \begin{flashcard}[Definition]{Norm on a Linear Space \\ Normed Space} A real-valued function $||x||$ defined on a linear space $X$, where $x \in X$, is said to be a \emph{norm on} $X$ if \smallskip \begin{description} \item [Positivity] $||x|| \geq 0$, \item [Triangle Inequality] $||x+y|| \leq ||x|| + ||y||$, \item [Homogeneity] $||\alpha x|| = |\alpha| \: ||x||$, $\alpha$ an arbitrary scalar, \item [Positive Definiteness] $||x|| = 0$ if and only if $x=0$, \end{description} \smallskip where $x$ and $y$ are arbitrary points in $X$. \medskip A linear/vector space with a norm is called a \emph{normed space}. \end{flashcard} \begin{flashcard}[Definition]{Inner Product} Let $X$ be a complex linear space. An \emph{inner product} on $X$ is a mapping that associates to each pair of vectors $x$, $y$ a scalar, denoted $(x,y)$, that satisfies the following properties: \medskip \begin{description} \item [Additivity] $(x+y,z) = (x,z) + (y,z)$, \item [Homogeneity] $(\alpha \: x, y) = \alpha (x,y)$, \item [Symmetry] $(x,y) = \overline{(y,x)}$, \item [Positive Definiteness] $(x,x) > 0$, when $x\neq0$. \end{description} \end{flashcard} \begin{flashcard}[Definition]{Linear Transformation/Operator} A transformation $L$ of (operator on) a linear space $X$ into a linear space $Y$, where $X$ and $Y$ have the same scalar field, is said to be a \emph{linear transformation (operator)} if \medskip \begin{enumerate} \item $L(\alpha x) = \alpha L(x), \forall x\in X$ and $\forall$ scalars $\alpha$, and \item $L(x_1 + x_2) = L(x_1) + L(x_2)$ for all $x_1,x_2 \in X$. \end{enumerate} \end{flashcard} \end{document} \endinput %% %% End of file `samplecards.tex'.