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chap6.log
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This is pdfTeX, Version 3.14159265-2.6-1.40.20 (MiKTeX 2.9.6960 64-bit) (preloaded format=pdflatex 2019.2.13) 8 APR 2019 19:36
entering extended mode
**"C:/Users/Bill Li/Desktop/MATH-110/chap6.tex"
("C:/Users/Bill Li/Desktop/MATH-110/chap6.tex"
LaTeX2e <2018-12-01>
! Undefined control sequence.
l.1 \section
{Inner Product Spaces}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! LaTeX Error: Missing \begin{document}.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.1 \section{I
nner Product Spaces}
You're in trouble here. Try typing <return> to proceed.
If that doesn't work, type X <return> to quit.
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.2 \subsubsection
*{Motivation}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Missing character: There is no * in font nullfont!
Missing character: There is no M in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
! LaTeX Error: Environment definition undefined.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.3 \begin{definition}
Your command was ignored.
Type I <command> <return> to replace it with another command,
or <return> to continue without it.
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
LaTeX Font Info: External font `cmex10' loaded for size
(Font) <7> on input line 4.
LaTeX Font Info: External font `cmex10' loaded for size
(Font) <5> on input line 4.
LaTeX Warning: Command \b invalid in math mode on input line 4.
LaTeX Warning: Command \b invalid in math mode on input line 4.
LaTeX Font Info: Try loading font information for +cmr on input line 4.
LaTeX Font Info: No file cmr.fd. on input line 4.
LaTeX Font Warning: Font shape `/cmr/m/n' undefined
(Font) using `/cmr/m/n' instead on input line 4.
! Corrupted NFSS tables.
wrong@fontshape ...message {Corrupted NFSS tables}
error@fontshape else let f...
l.4 In $\b R
^n$, the dot product of $\vec x$ and $\vec y$ is defined by
This error message was generated by an \errmessage
command, so I can't give any explicit help.
Pretend that you're Hercule Poirot: Examine all clues,
and deduce the truth by order and method.
LaTeX Font Warning: Font shape `/cmr/m/n' undefined
(Font) using `OT1/cmr/m/n' instead on input line 4.
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Overfull \hbox (20.0pt too wide) in paragraph at lines 1--5
[]
[]
Overfull \hbox (12.80444pt too wide) in paragraph at lines 1--5
[][]$
[]
Overfull \hbox (5.71527pt too wide) in paragraph at lines 1--5
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Overfull \hbox (5.2616pt too wide) in paragraph at lines 1--5
[]$
[]
Overfull \hbox (125.55112pt too wide) detected at line 5
[] \OMS/cmsy/m/n/10 [] \OT1/cmr/m/n/10 := \OML/cmm/m/it/10 x[]y[] \OT1/cmr/m/
n/10 + \OML/cmm/m/it/10 x[]y[] \OT1/cmr/m/n/10 + [] + \OML/cmm/m/it/10 x[]y[]
[]
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
! Undefined control sequence.
l.6 for $\vec x = (\li
xn), \vec y = (\li yn)$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.6 for $\vec x = (\li xn), \vec y = (\li
yn)$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Missing character: There is no . in font nullfont!
! LaTeX Error: \begin{document} ended by \end{definition}.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.7 \end{definition}
Your command was ignored.
Type I <command> <return> to replace it with another command,
or <return> to continue without it.
! Undefined control sequence.
l.8 \subsection
{Inner Product and Norms}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no P in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.9 \subsubsection
*{Settings}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Missing character: There is no * in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
LaTeX Warning: Command \b invalid in math mode on input line 10.
LaTeX Warning: Command \b invalid in math mode on input line 10.
Missing character: There is no , in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
! Undefined control sequence.
l.10 ... we can define the following mapping $ \la
\cdot, \cdot \ra : V \tim...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.10 ...e following mapping $ \la \cdot, \cdot \ra
: V \times V \to \b F$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
LaTeX Warning: Command \b invalid in math mode on input line 10.
LaTeX Warning: Command \b invalid in math mode on input line 10.
Missing character: There is no . in font nullfont!
! LaTeX Error: Environment definition undefined.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.11 \begin{definition}
Your command was ignored.
Type I <command> <return> to replace it with another command,
or <return> to continue without it.
! Undefined control sequence.
<recently read> \la
l.12 $\la
\cdot, \cdot \ra$ is called an inner product if it satisfying the ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.12 $\la \cdot, \cdot \ra
$ is called an inner product if it satisfying the ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no : in font nullfont!
Overfull \hbox (16.27078pt too wide) in paragraph at lines 5--13
$[] \/cmr/m/n/10 =
[]
Overfull \hbox (39.75694pt too wide) in paragraph at lines 5--13
\/cmr/m/n/10 (\OML/cmm/m/it/10 xn\/cmr/m/n/10 )\OML/cmm/m/it/10 ; [] \/cmr/m/n/
10 =
[]
Overfull \hbox (19.04175pt too wide) in paragraph at lines 5--13
\/cmr/m/n/10 (\OML/cmm/m/it/10 yn\/cmr/m/n/10 )$
[]
Overfull \hbox (8.05556pt too wide) in paragraph at lines 5--13
\OML/cmm/m/it/10 V$
[]
Overfull \hbox (6.5278pt too wide) in paragraph at lines 5--13
[]$
[]
Overfull \hbox (15.55548pt too wide) in paragraph at lines 5--13
\OMS/cmsy/m/n/10 \OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 \/cmr/m/n/10 :
[]
Overfull \hbox (15.83336pt too wide) in paragraph at lines 5--13
\OML/cmm/m/it/10 V \OMS/cmsy/m/n/10
[]
Overfull \hbox (20.83328pt too wide) in paragraph at lines 5--13
\OML/cmm/m/it/10 V \OMS/cmsy/m/n/10 !
[]
Overfull \hbox (6.5278pt too wide) in paragraph at lines 5--13
[]$
[]
Overfull \hbox (9.99998pt too wide) in paragraph at lines 5--13
\OMS/cmsy/m/n/10 \OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 $
[]
Missing character: There is no ( in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ) in font nullfont!
! Undefined control sequence.
<recently read> \la
l.14 \item(additivity in the first slot) $\la
\vec v + \vec u, \vec w \r...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.14 ...rst slot) $\la \vec v + \vec u, \vec w \ra
= \la \vec v + \vec w \ra...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.14 ...ot) $\la \vec v + \vec u, \vec w \ra = \la
\vec v + \vec w \ra + \la...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.14 ...ec u, \vec w \ra = \la \vec v + \vec w \ra
+ \la \vec u, \vec w \ra,...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.14 ...\vec w \ra = \la \vec v + \vec w \ra + \la
\vec u, \vec w \ra, \ \fo...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.14 ...ec v + \vec w \ra + \la \vec u, \vec w \ra
, \ \forall \vec v, \vec u...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Overfull \hbox (12.98381pt too wide) in paragraph at lines 14--15
[] \/cmr/m/n/10 +
[]
Overfull \hbox (28.15276pt too wide) in paragraph at lines 14--15
[]\OML/cmm/m/it/10 ; [] \/cmr/m/n/10 =
[]
Overfull \hbox (12.98381pt too wide) in paragraph at lines 14--15
[] \/cmr/m/n/10 +
[]
Overfull \hbox (15.20605pt too wide) in paragraph at lines 14--15
[] \/cmr/m/n/10 +
[]
Overfull \hbox (64.28929pt too wide) in paragraph at lines 14--15
[]\OML/cmm/m/it/10 ; []; \OMS/cmsy/m/n/10 8[]\OML/cmm/m/it/10 ; []; [] \OMS/cm
sy/m/n/10 2
[]
Overfull \hbox (8.05556pt too wide) in paragraph at lines 14--15
\OML/cmm/m/it/10 V$
[]
Missing character: There is no ( in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ) in font nullfont!
! Undefined control sequence.
<recently read> \la
l.15 \item(homogeneity in the first slot) $\la
\lambda \vec v, \vec w \r...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.15 ...irst slot) $\la \lambda \vec v, \vec w \ra
= \lambda \la \vec v + \v...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.15 ...a \lambda \vec v, \vec w \ra = \lambda \la
\vec v + \vec w \ra, \ \f...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.15 ...ec w \ra = \lambda \la \vec v + \vec w \ra
, \ \forall \vec v \vec u,...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
LaTeX Warning: Command \b invalid in math mode on input line 15.
LaTeX Warning: Command \b invalid in math mode on input line 15.
Overfull \hbox (33.46754pt too wide) in paragraph at lines 15--16
\OML/cmm/m/it/10 []; [] \/cmr/m/n/10 =
[]
Overfull \hbox (18.81717pt too wide) in paragraph at lines 15--16
\OML/cmm/m/it/10 [] \/cmr/m/n/10 +
[]
Overfull \hbox (49.67589pt too wide) in paragraph at lines 15--16
[]\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8[][]\OML/cmm/m/it/10 ; [] \OMS/cmsy/m/n
/10 2
[]
Overfull \hbox (26.11105pt too wide) in paragraph at lines 15--16
\OML/cmm/m/it/10 V; \OMS/cmsy/m/n/10 2
[]
Overfull \hbox (6.5278pt too wide) in paragraph at lines 15--16
[]$
[]
Missing character: There is no ( in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no ) in font nullfont!
! Undefined control sequence.
<recently read> \la
l.16 \item(conjugate symmetry) $\la
\vec v, \vec w \ra = \overline{\la \...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.16 ...onjugate symmetry) $\la \vec v, \vec w \ra
= \overline{\la \vec w, \...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.16 ...y) $\la \vec v, \vec w \ra = \overline{\la
\vec w, \vec v \ra}, \ \f...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.16 ...c w \ra = \overline{\la \vec w, \vec v \ra
}, \ \forall \vec v,\vec w...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Overfull \hbox (27.63419pt too wide) in paragraph at lines 16--17
[]\OML/cmm/m/it/10 ; [] \/cmr/m/n/10 =
[]
Overfull \hbox (53.60173pt too wide) in paragraph at lines 16--17
[]\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8[]\OML/cmm/m/it/10 ; [] \OMS/cmsy/m/n/1
0 2
[]
Overfull \hbox (8.05556pt too wide) in paragraph at lines 16--17
\OML/cmm/m/it/10 V$
[]
Missing character: There is no ( in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no ) in font nullfont!
! Undefined control sequence.
<recently read> \la
l.17 \item(positivity) $\la
\vec v, \vec v \ra \geq 0, \ \forall \vec v ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.17 \item(positivity) $\la \vec v, \vec v \ra
\geq 0, \ \forall \vec v ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Overfull \hbox (25.41194pt too wide) in paragraph at lines 17--18
[]\OML/cmm/m/it/10 ; [] \OMS/cmsy/m/n/10
[]
Overfull \hbox (29.6504pt too wide) in paragraph at lines 17--18
\/cmr/m/n/10 0\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 8[] 2
[]
Overfull \hbox (8.05556pt too wide) in paragraph at lines 17--18
\OML/cmm/m/it/10 V$
[]
Missing character: There is no ( in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no ) in font nullfont!
! Undefined control sequence.
<recently read> \la
l.18 \item(definiteness) $\la
\vec v, \vec v \ra = 0$ iff $\vec v = \vec...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.18 ...item(definiteness) $\la \vec v, \vec v \ra
= 0$ iff $\vec v = \vec 0$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no . in font nullfont!
Overfull \hbox (25.41194pt too wide) in paragraph at lines 18--19
[]\OML/cmm/m/it/10 ; [] \/cmr/m/n/10 =
[]
Overfull \hbox (5.00002pt too wide) in paragraph at lines 18--19
\/cmr/m/n/10 0$
[]
Overfull \hbox (15.76152pt too wide) in paragraph at lines 18--19
[] \/cmr/m/n/10 =
[]
Overfull \hbox (5.00002pt too wide) in paragraph at lines 18--19
[]$
[]
! LaTeX Error: \begin{document} ended by \end{definition}.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.20 \end{definition}
Your command was ignored.
Type I <command> <return> to replace it with another command,
or <return> to continue without it.
! LaTeX Error: Environment question undefined.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.21 \begin{question}
Your command was ignored.
Type I <command> <return> to replace it with another command,
or <return> to continue without it.
Missing character: There is no W in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ? in font nullfont!
! LaTeX Error: \begin{document} ended by \end{question}.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.23 \end{question}
Your command was ignored.
Type I <command> <return> to replace it with another command,
or <return> to continue without it.
! LaTeX Error: Environment answer undefined.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.24 \begin{answer}
We can compute
Your command was ignored.
Type I <command> <return> to replace it with another command,
or <return> to continue without it.
Missing character: There is no W in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.25 \[ \la
\vec v, \vec u + \vec w \ra = \overline{\la \vec u + \vec w, \ve...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.25 \[ \la \vec v, \vec u + \vec w \ra
= \overline{\la \vec u + \vec w, \ve...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.25 ...vec v, \vec u + \vec w \ra = \overline{\la
\vec u + \vec w, \vec v\r...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.25 ...= \overline{\la \vec u + \vec w, \vec v\ra
} = \overline{\la \vec u, ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.25 ...vec u + \vec w, \vec v\ra} = \overline{\la
\vec u, \vec v \ra + \ra ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
! Undefined control sequence.
l.25 ...c v\ra} = \overline{\la \vec u, \vec v \ra
+ \ra \vec w, \vec v} = \...
The control sequence at the end of the top line