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(Definition)
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Let
be either
or
, and let
with . We define to be the set of all sequences
in
such that
converges.
We also define
to be the set of all bounded sequences
with norm given by
By defining addition and scalar multiplication pointwise,
and
have a natural vector space stucture. That the sum of two elements on
is again an element in
follows from Minkowski inequality (see below). We can make into a normed vector space, by defining the norm as
The normed vector spaces
and for are complete under these norms, making them into Banach spaces. Moreover, is a Hilbert space under the inner product
where
denotes the complex conjugate of .
For the (continuous) dual space of is where
, and the dual space of is
.
- If
for
, then
. (proof.)
- For
,
is separable, and
is not separable.
- Minkowski inequality. If
where , then
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" " is owned by rspuzio. [ full author list (3) | owner history (1) ]
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(view preamble)
See Also: space
| Also defines: |
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Cross-references: separable, dual space, continuous, complex conjugate, inner product, Hilbert space, Banach spaces, complete, normed vector space, Minkowski inequality, sum, vector space, pointwise, multiplication, scalar, addition, norm, converges, sequences
There are 2 references to this entry.
This is version 19 of , born on 2002-02-13, modified 2005-05-21.
Object id is 1929, canonical name is Lp.
Accessed 5376 times total.
Classification:
| AMS MSC: | 46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous) | | | 54E50 (General topology :: Spaces with richer structures :: Complete metric spaces) |
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Pending Errata and Addenda
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