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Definition 1 Vector space is a non empty set V if the following are met:
- An operation, which will be called vector addition and denoted as +, is defined between any two vectors in V in such a way that if u and v are in V, then u+ v is too (i.e., V is closed under addition). Furthermore,
(u+ u) + w = u+ (v+ w). (associative) |
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- V contains a unique zero vector 0 such that
for each u in V.
- For each u in V there is a unique vector "-u" in V, called the negative inverse of u, such that
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- Another operation, called scalar multiplication, is defined such that if u is any vector in V and a is any scalar, then the scalar multiple au is in V, too (i.e., V is closed under scalar multiplication). Further, we require that
a(bu) = (ab) u (associative) |
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(a+b) u = au+bu (distributive) |
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a( u+v) = au+av (distributive) |
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if the vectors u, v are in V, and a,b are scalars.
Definition 2 vector space H is called an inner product space if to each pair of vectors u and v in H is associated a number (u,v) such that the following rules hold:
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Definition 3 vector space V is called an norm space if to each vector u in V is associated a number ||u|| such that the following rules hold:
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||u+v|| £ ||u||+||v|| (triagleinequality) |
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File translated from TEX by TTH, version 2.25.
On 29 Mar 2001, 13:50.