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Consider the term "functional operator". My understanding was that:

(a) An operator in this context refers to a mapping from one vector space to another vector space.

(b) A functional is a mapping from a vector space to its underlying scalar field.

(c) Scalars are fundamentally different objects than vectors, and cannot, for instance, be thought of as little 1x1 vectors.

So if a functional is a mapping from a vector space into a scalar field, how can it be an operator? And I know that "operator" can sometimes have a much more general meaning ("something that does something") but I'm pretty sure in the context of a phrase like "functional operator" we're clearly in the more specific conversation about vector spaces.

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    ... the scalars form a vector space ...2012-07-20
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    I think point (c) is off the mark. Any field satisfies the axioms of a vector space over that field. There are some instances, like Calc III where we make a big deal of distinguishing vectors from scalars, but what we mean in that context is that $1$-vectors should not be confused with $2$-vectors or $3$-vectors.2012-07-20
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    thanks guys, this is helpful! but would you mind fleshing it out a little more for me? I thought a scalar was just a magnitude, while a vector has more structure - it's a magnitude and a set of directions. Is a scalar field (a collection of scalars) really a kind of vector space (a collection of vectors)? Even if the scalars can be brought together to form vectors, does this really imply that a scalar field is vector space, instead of the ingredients for a vector space? And would framing this in terms of tensors help to understand how vectors and scalars are part of the same larger framework?2012-07-20
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    @Lucas The general notion of a vector is somewhat different. A vector is simply an element of a vector space, which can be very strange objects. For example, $\mathbb R$ is an infinite-dimensional vector space over $\mathbb Q$.2012-07-20
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    Do you know the definition of a vector space?2012-07-20
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    (B) is the definition of "functional (noun)". The expression in the title involves "functional (adjective)". Dictionaries teach us that the meaning if two can be different.2012-07-21

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