Let V be the set of all real numbers. Define an operation of "addition" by xy = the maximum of x and y for all x, y € V. Define an operation of "scalar multiplication" by a x = ax for all a E R and x € V. Under the operations and the set V is not a vector space. The vector space axioms (see 5.1.1 (1)-(10)) which fail to hold are and "

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