Reciprocal lattice in crystallography Where do the formulas for reciprocal lattice vectors come from? 
I recently began studying tensors and the same formula's appeared yet again, this time called dual basis vectors! This reignited my interest in a derivation of the reciprocal lattice. 
Can someone provide some insight into this?
 A: Caution: I am too lazy to edit everything that comes below. But my friend has pointed out that vector spaces are defined over a field and integers don't form a field. So the reciprocal lattice vectors don't span a vector space(nor do the direct vectors). Just keep that in mind as you read what follows.
Disclaimer: This answer may or may not be right. When I read your question, it felt like a lot of puzzle pieces had fallen into place. Suddenly a lot of things that I studied in two different courses made total sense. So I am writing this answer, but it would be very nice if someone could tell us if I am right.
First, I'll tell you what dual vectors really are (I was surprised when I found this out). Then I'll tell you how it totally makes so much sense to see the reciprocal lattice vectors as the dual vectors of the direct lattice vectors. 
What is a dual vector? A dual vector is a map that takes a vector to a number (real number if your vector space is a real vector space). So say $\vec{V}$ is a vector in a real vector space, then a dual vector $\omega$ is: $$\omega: \vec{V} \rightarrow \mathbb{R}$$ This is not easy to digest. You say, "I used to think that the dual vectors are also vectors. And now you say that they are functions? How am I even supposed to take a dot product between a dual vector a vector when the dual vector is a function?" At this point you have to realize that functions can form a vector space. And if they form a vector space then you can choose a basis for them. So now you have two vector spaces, one (say $V$) to which the vectors belong and the other (say $D$) to which the dual vectors belong. Let us also assume that we have chosen the bases for these spaces as {$\hat{e}^{\mu}$} for $V$ and {$\hat{k}_{\nu}$} for $D$. So, we have for any vector and dual vector : $$\vec{v} = \sum c_i \hat{e}^i \in V $$ $$\vec{\omega} = \sum d_i \hat{k}_i \in D $$
Notice that the index for one basis is written as a superscript and the other as a subscript. This is done to emphasize the fact that these tho entities are both vectors, but they are very different kinds of vectors. So we have to define how these two entirely different (but similar) beasts interact with each other. To that end let us choose the action of the dual vector $\hat{k}_j$ on the vector $
\hat{e}^{\mu}$: $$ \hat{e}^i \cdot \hat{k}_j = \delta_{j}^i$$ Notice that this is consistent with the first equation and the dual vector is mapping the vector to a real number that is either 0 or 1. If these are satisfied then the action of the dual vector on the vector is given by the expreession: $$\vec{v} \cdot \vec{\omega} = \left(\sum c_i \hat{e}^i \right)\cdot \left(\sum d_j \hat{k}_j\right) = \sum c_i d_j \delta_j^i = \sum c_i d_i$$ Thus you recover the definition of the inner product. This is also why you can write the dual vectors and vectors a collection of components. As long as the action of the basis dual vectors on the basis vectors is defined, you only need the components of the vectors to take the inner product.
Now, we come to the second part. Why are the reciprocal lattice vectors just dual vectors? Don't you see already? The reciprocal lattice vectors map the vectors in the direct lattice to real numbers. Therefore they satisfy the first equations and are dual vectors. Now the next question it raises is how are these linear maps defined? So basically we need the equivalent of the fourth equation above. I am not absolutely sure about this but I think this might work: $$\hat{e}^i \cdot \hat{k}_j = 2\pi\delta_{j}^i$$  Since the components of the direct lattice vectors and reciprocal lattice vectors can only be integers then their components will only multiply an integer to $2\pi$, so say $\vec{a}$ is a direct lattice vector and $\vec{b}$ is a reciprocal lattice vector then the action of $\vec{b}$ on $\vec{a}$ is: $$\vec{b}\cdot\vec{a}=\left(\sum a_i \hat{e}^i \right)\cdot\left(\sum b_j \hat{k}_j\right) = \sum a_i b_j \hat{e}^i \cdot \hat{k}_j = 2\pi \sum a_i b_i$$ Since the components of the direct and reciprocal lattice vectors can only be integers we have the result that this is equal to $2 \pi m$ where $m$ is an integer. And there you have it! The definition of reciprocal lattice vectors.
A: In the end they are just a very powerful and helpful definition. The reciprocal vectors show up in many important crystallographic equations. For example the Laue law:
$$\mathbf{k}_{\rm in} - \mathbf{k}_{\rm out} = \mathbf{G}_{hkl}\\
G_{hkl} = h \mathbf{b}_1 +k \mathbf{b}_2 + l\mathbf{b}_3 $$
With the Laue law you can find allowed x-ray reflections. Also the distance between 2 consecutive planes with Miller indice hkl is given by
$$d_{hkl} = \frac{2\pi}{|\mathbf{G}_{hkl}|}$$
