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Norms of Vectors
When we want to measure the length of, or distance between, vectors we need a
yardstick that measures in a consistent way, generalizing the idea of absolute value
to vector spaces. We can capture the essence of length by requiring that such a
yardstick satisfy the following properties:
1.
f
(
x
)
>
0
,
∀
x
6
= 0,
2.
f
(
αx
) =

α

f
(
x
)
,
∀
x
and for all scalars
α
, and
3.
f
(
x
+
y
)
≤
f
(
x
) +
f
(
y
)
,
∀
x
and
y
.
Any function which satisﬁes these properties can be called a
norm
, and we usually
use the symbol
f
(
x
) =
k
x
k
. With a norm,
R
n
is a metric space, with the distance
between
x
and
y
being
d
(
x, y
) =
k
x

y
k
. The
unit circle
or
unit ball
in (
R
n
,
k · k
) is
the set of vectors with unit length:
{
x
∈
R
n
:
k
x
k
= 1
}
. A norm is completely
determined by its unit circle (property 2), so don’t be suprised if almost all unit
circles aren’t.
.. circular.
You have already seen the Euclidean norm, it is also called the 2norm because it is
the
p
= 2 member of this family of pnorms:
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 Fall '10
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