Suppose I have a set involving trigonometric functions, with addition defined as multiplication of two vectors. If this is a vector space, the zero vector has to be unique. If cos (0) works as the zero vector, then cos (2*pi), etc. also work. Does this mean the set is not a vector space, because the zero element is not unique? Or is it still a vector space (all other axioms check out) because cos (0) = cos (2*pi) = 1 ?(adsbygoogle = window.adsbygoogle || []).push({});

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# Question about zero element in vector spaces

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