Motivation behind Definition of Inner Product Space What is the motivation behind definition of inner product space?Why mathematicians gave definition of inner product in this way not any other way?What is the reason?
 A: Originally, the concept of a vector space arose out of quaternions. William Rowan Hamilton had been impressed about how complex numbers could be used to develop and discuss planar geometry. For many years he attempted to figure out a similar 3-dimensional algebra. Finally he realized that 4 dimensions were required, with three imaginary units $i, j, k$, which satisfied the equations $$ij = -ji = k,\quad jk = -kj = i,\quad ki = -ik = j\\i^2 = j^2 = k^2 = -1$$
Famously, he realized this on a walk with his wife, and was so excited, he carved these equations into the wood of a bridge they were passing over.
Similar to complex numbers, a quaternion has the form $a + bi + cj + dk$ for real numbers $a, b, c, d$. Hamilton labeled $a$ the scalar part and $bi + cj + dk$ the vector part, or just "scalar" and "vector", introducing both terms into mathematics. He identified the vector part with points in space and for a while they were the way to do analytic geometry.
But later developers, in particular Heaviside, simplified his descriptions and methods, and realized that pretty much everything that they wanted to do could be done just with the vector part. The scalar term was not necessary. This lead them to formalize the concept of a vector space as we know it today.
So what about the inner product? If you multiply two quaternion vectors together, you get another quaternion with both scalar and vector parts:
$$(ai+bj+ck)(xi + yj + zk) = -(ax + by + cz) + (bz - cy)i + (cx - az)j + (ay - bx)k$$
Both the scalar part (or more particularly, its opposite) and vector part were very useful in discussing geometry. Heaviside gave them the names "dot product" and "cross product", respectively, after the symbols "$\cdot$" and "$\times$" he used to denote them.
The name "inner product" was introduced when these 3D vectors became identified with $3\times 1$ column matrices, where matrix multiplication provides two products for vectors: the "inner product" $v^Tw$ is a $1\times 1$ matrix, identifiable with scalar number, and is the same as the dot product, while the similar "outer product" $vw^T$ gives a $3 \times 3$ matrix, that corresponds to a projection operator onto the line of $v$ (when $w = v$ and $v$ is a unit vector, it is the orthogonal projection onto the line).
When vector spaces were first generalized to $\Bbb R^n$ for arbitrary $n$, the inner product formula was obviously generalizable to any dimension. (The cross product also has a fairly obvious generalization, except it goes from being a binary operation in 3-dimensional space to being an $(n-1)$-ary operation in $n$-dimensional space, which limits its usefulness.) This generalized inner product kept its strong geometric interpretation (which I expect you know), and is very useful in any dimension.
When vector spaces were generalized further to its modern form, the natural basis of $\Bbb R^n$ disappeared, and without it, the coefficient formula could no longer be used to define the inner product. So arbitrary vector spaces do not come with an automatic inner product. Instead, mathematicians identified the properties of the $\Bbb R^n$ inner product that were useful, and defined that an inner product on an arbitrary vector space should still be required to meet them.
If you are wondering about why complex inner products must satisfy $$\langle u, v\rangle = \overline {\langle v, u\rangle}$$
This comes from the need for the norm $$\sqrt{\langle u, u\rangle}$$ to be a real number, while still satisfying a form of bilinearity. And the norm needs to be real so it can define a distance function and those distances can be compared to each other, and thus used to define a topology (i.e., limits & continuity).
A: I myself only relatively recently learned that the notions of "vector space" and "inner product" were most vigorously developed in (not finite-dimensional) spaces of functions, from the time of Fourier and Dirichlet, Sturm and Liouville, and so on, throughout the 19th century. The notion of "inner product" arose in Fourier's work, as the device to determine Fourier series coefficients. A similar device existed in Sturm-Liouville theory.
It is my impression that, in fact, this thread played a larger role than the quaternion thread of Hamilton and Gibbs, and therefore was much more developed. Indeed, Hilbert, Schmidt, Volterra, and others developed quite sophisticated ideas about compact self-adjoint operators on Hilbert spaces and more, pre-1900. As far as I can tell, in fact this infinite-dimensional context guided the finite-dimensional, contrary to a "logical" development. After all, the need for the infinite-dimensional case was greater, given the questions of interest of the time.
An example of an issue that mattered greatly at the time (late 19th century, early 20th) is the fact that the original and several other formulations of "Dirichlet's principle" were patently false, although the conclusions that intuitive people (Riemann, et al) reached were true. The situation was not clarified until B. Levi and Frobenius reframed the issue in certain Hilbert spaces, later called Sobolev spaces, proving a (true!) version of the Dirichlet principle, and recovering the necessary (true) results. The geometry of Hilbert spaces is/was essential in this.
