[Corpora-List] QM analogy and grammatical incompleteness
John Williams
johnwhoever at wanadoo.fr
Tue Dec 20 09:26:00 CET 2005
> In broad strokes, the history of vectors and functional
analysis became
> very closely linked in the 1840s and 1850s, partly through
Hamilton's
> work on quaternions and the theory of analytic functions on
4-space.
> Functions over the real numbers form a vector space - you can
add two
> functions together, and multiply any function by a scalar. As
a result,
> mathematicians came to realize that Fourier analysis could be
described
> in vectors - each of the functions sin(nx) and cos(nx) (for x
a real
> number, n an integer) is a basis vector, and any piecewise
smooth
> function can be expanded (uniquely) as a vector, using these
functions
> as a basis. The Fourier series coefficients are thus
interpreted as the
> coordinates of a vector in this basis. This vector space is
clearly
> infinite-dimensional, because a Fourier series expansion can
be
> infinitely long. (Note again that this means you will never
work with
> complete information once you've quantized your functions.)
And I took the languages option at school so I wouldn't have to do maths
and physics. Sigh.
Merry Christmas everybody.
John Williams
--
John Williams
Sometime Corpus Lexicographer and English Teacher.
17 rue Thionville
31000 TOULOUSE
France
Tel: (+33) (0)5 61 99 03 86
Mob: (+33) (0)6 76 12 42 24
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