1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
|
%%
%% 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 <ambudge@mit.edu>
%%
%% 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'.
|
