Mathematical Methods for Physicists: A concise introduction - Site Map

DUAL VECTORS AND DUAL SPACES
or
jhUjWij jUjjWj;
which is the Cauchy±Schwarz inequality.
From the Cauchy±Schwarz inequality follows another important inequality,
known as the triangle inequality,
jU ‡ W
j jUj‡ jWj: …5:14†
The proof of this is very straightforward. For any pair of vectors, we have
jU ‡ W
j 2ˆ hU ‡ W
from which it follows that
jU ‡ Wi ˆ jU
jU ‡ W
jU
jU
j 2 ‡ jW
j 2 ‡ jW
j 2 ‡ jW
j 2 ‡ hU
j jUj‡ jWj:
jWi ‡ hWjUi
j 2 ‡ 2jhUjWij
j 2 ‡ 2jU
jjW
j… jUj 2 ‡ jWj 2 †
If V denotes the vector space of real continuous functions on the interval
a x b, and f and g are any real continuous functions, then the following is
an inner product on V:
hf
jgi ˆ
Z b
a
f …x†g…x†dx:
The Cauchy±Schwarz inequality now gives
Z b
2 Z b
f …x†g…x†dx f 2 …x†dx
or in Dirac notation
a
jhf
jgij 2 jf
j 2 jgj 2 :
a
Z b
a
g 2 …x†dx
Dual vectors and dual spaces
We begin with a technical point regarding the inner product hujvi. If we set
jvi ˆjwi‡jzi;
then
hujvi ˆhujwi‡hujzi
is a linear function of and . However, if we set
jui ˆjwi‡jzi;
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