If the tangent space at p is a true vector space, then it must be that the sum of two vectors is itself a directional derivative operator along some path passing through p. I've been trying to prove that this is true without any luck.(adsbygoogle = window.adsbygoogle || []).push({});

My textbook "proves" that vector addition is closed by showing that when the sum of two vectors is applied to the product of two scalar functions, we just get the Leibniz rule. This argument seems really unconvincing to me.

Can anyone either justify the validity of my text's proof, or offer their own proof for the closure of vector addition?

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# Closure of vector addition

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