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Vector Spaces and Hilbert Spaces
( Mainly, Griffiths A.1, A.2)
A
vector space
is a set of objects called vectors ( 
A
⟩
, 
B
⟩
, 
C
⟩
,...) and a set of numbers called scalars
(a, b, c,.
..) along with a rule for vector addition and a rule for scalar multiplication. If the scalars are real,
we have a
real vector space
; if the scalars are complex, we have a
complex vector space
. The set must
be
closed
under vector addition and scalar multiplication.
Vector addition
must have these properties:
•
The sum of any 2 vectors is a vector:

A
⟩
+ 
B
⟩
= 
C
⟩
•
Vector addition is commutative and associative:

A
⟩
+ 
B
⟩
= 
B
⟩
+ 
A
⟩
and

A
⟩
+ (
B
⟩
+ 
C
⟩
) = (
A
⟩
+
B
⟩
) +
C
⟩
•
There exists a zero vector 
0
⟩
such that :

A
⟩
+ 
0
⟩
= A
⟩
for any vector 
A
⟩
•
For every vector 
A
⟩
there is an inverse vector 
A
⟩
such that 
A
⟩
+

A
⟩
=

0
⟩
Scalar multiplication
must have these properties:
•
The product of a scalar and a vector is another vector:
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 Fall '08
 STEVEPOLLOCK

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