# undetstanding of Vectors, Tangent Vectors, Tangent Covectors

To my limited knowledge, I only know vector as a certain fixed number of real numbers put together. for example $[1,2.3,6.4,0.75]$ is a vector. A vector of dimension $N$ is any of the elements of the set $\underbrace{\mathbb{R} \times \mathbb{R}\times.....\mathbb{R}}_{N \text{ times}}$. Ok, I also know the that sequences and functions of a real variable can also be vectors, for example the collection of all square integrable functions or the collection of all square summable sequences can be vector spaces. I can imagine the notion of addition of vectors in all these examples as addition of corresponding elements of two vectors to form the corresponding element of the sum vector, for example the addition of two square integrable functions $f$ and $g$ is nothing but the pointwise addition of values of the function to form the values of the resultant sum function $f+g$.

The point-wise addition is a must for me to imagine a vector. But I am finding hard and clueless when I try to read and understand concepts like tangent vectors and tangent co-vectors. I am clueless and I can't even try to explain my difficulty, hope someone understands my problem and put things for me so that i can overcome this. I was also reading this answer by Aaron here but i am very far from understanding things like "These tangent vectors act on functions by taking the directional derivative of a function at a point. If you take a tangent covector, it no longer acts on functions, it just acts on vectors. " and "a "dual" space V∗ which consists of linear functions V→𝔽 (where 𝔽 is the underlying field)." I do not understand how linear functions V→𝔽 can be called as vectors. I can go to the extent of reading references but i failed a few times and need some advice.

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