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Definition 1 Vector space is a non empty set V if the following are met:

1. 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+v = v+u (commutative)

   (u+ u) + w = u+ (v+ w). (associative)

2. V contains a unique zero vector 0 such that

   u+ 0 = u

   for each u in V.

3. For each u in V there is a unique vector "-u" in V, called the negative inverse of u, such that

   u+ (-u) = 0

   .

4. 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)

   (a+b) u = au+bu (distributive)

   a( u+v) = au+av (distributive)

   lu = u

   ,

   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:

1. (u,v) = (v,u)

2. (u+w,v) = (u,v)+(w,v)

3. (au,v) = a(u,v)

4. (u,u) ’†0

5. (u,u) = 0 iff u = 0

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:

1. ||u+v|| £ ||u||+||v|| (triagleinequality)

2. ||au|| = |a|||u||

3. || u|| = 0 iff u = 0