VectorSpace - Vector Spaces and Hilbert Spaces ( Mainly,...

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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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