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For an arbitrary vector space one defines, the scalar field can be anything: real, complex, rational, or some other previously defined field.

However, in some texts on linear algebra for example, the authors do not define what a scalar is. Is it conventional to assume that, unless specified otherwise, a scalar is a real number? If not, in what topics, if any, is the conventional scalar taken to be complex numbers or some other field?

I realize this question may be somewhat fluffy, but I'd appreciate any inputs you may have.

TSJ
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  • In order to be perfectly unambiguous, one would (and should) say things like "Let $C^\infty(\Bbb R)$ be the vector space over the reals of all continuous functions $\Bbb R\to\Bbb R$" in order to emphasize what the scalar field is. – JMoravitz Jul 27 '16 at 04:19

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A scalar is simply an element of the field where the vector space is defined over.

It is usually understood what field you are working over.

So if V is a k-vector space, then a scalar is an element of k.

LASV
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