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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