Typically an element of a vector space is called a vector, but Carroll's GR book repeatedly refers to elements of tangent spaces as "transforming as a vector" when they change coordinates as V(adsbygoogle = window.adsbygoogle || []).push({}); ^{μ}= ∂x^{μ}/∂x^{ν}V^{ν}. However, dual vectors are members of vector spaces (cotangent space) but obey ω_{μ}= ∂x^{v}/∂x^{μ}ω_{v}. Is this abuse of terminology? If so, what is a more exact way of describing objects in vector spaces obeying the vector transformation law?

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# I Vector Transformation Law and Vector Spaces

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