Intuitive Explanation of the Inner Product I'm currently studying Linear Algebra, and we've arrived at the topic of Inner Products and orthogonality. I have been searching online to try and help myself understand what the inner product actually is. I understand it is supposedly an generalisation of the dot product, and I understood that the dot product was useful in calculating the angle between two vectors in 3D space, etc. 
What is the inner product of two elements of real/ complex spaces? Is it a formula? What is the use of the inner product, and can it be applied to any vector space or specifically the real/ complex spaces? How does this idea extend to inner product spaces? I don't understand it intuitively and any help/ guidance on how to grasp the concept better would be really appreciated. Thanks.
 A: Inner products arise in a variety of areas of mathematics. In particular, vector spaces with inner products defined on it are usually called $\textit{inner product spaces}$, and this is very important when studying functional analysis. For vectors in $\mathbb{R}^{d}$, for example, an inner product can be defined as follows: if $x=(x_{1},...,x_{n}),y=(y_{1},...,y_{n})\in \mathbb{R}^{n}$ we set $$<x,y>:= \sum_{i=1}^{n}x_{i}y_{i}$$
Note that this is not the only possible inner product in $\mathbb{R}^{n}$. It may occur that a vector space has more than one inner product. See this discussion, for example. 
There are a lot of "good properties" that inner product spaces share and one of the most important properties is orthogonality. This is very commonly used, for instance, not just in mathematics but in physics and other applied sciences.
 Another very important (and interesting) property is that once you have an inner product on a vector space, you can readily define a $\textit{norm}$ from it. For instance, we can define a norm $||\cdot||$ in $\mathbb{R}^{d}$ by setting $$||x|| := \sqrt{<x,x>}= \sqrt{\sum_{i=1}^{n}x_{i}^{2}}$$
This very definition works in general, even if we are dealing with more abstract spaces. Vector spaces which have a norm defined on it are called $\textit{normed spaces}$. These spaces are studied in functional analysis courses as well. 
It is worth mentioning that Hilbert and Banach spaces (you may also have heard about these) are special cases of inner product spaces and normed spaces, respectivelly. 
As an application, I can say that inner products (and Hilbert spaces, in particular) are very important in quantum mechanics, for example. In short, this is because different states of a system are orthogonal, so the notion of orthogonality plays an important role on the theory.
