Dear Deepthi,

the concept of inner product is basic for Euclidean and Hilbert spaces. Inner product allows to consider lengths of vectors and angle between two vectors. In particular, one can speak about orthogonal vectors. Thus, inner product gives rise to metric  properties of a vector space.

You can also see Remarks in

http://www.math.harvard.edu/~knill/teaching/summer2011/handouts/12-dotproduct.pdf

Inner product is also a special bilinear form, see

http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/bilinearform.pdf

Best regards, Yuri

Addendum. If the basic field is C, the field of complex numbers, then an inner product is a sesquilinear form. It is bilinear if the basic field is R :)

In a real or complex vector space, if a product of vectors is defined satisfying certain conditions ( is non negative and =0 if and only if v=0 etc.), then the vector space is called real or complex inner product space. Here the  notation < ., .> is used to denote the product of two vectors.  Euclidean space is an example of an inner product space. 

If the inner product space is complete under the norm induced by the inner product i.e., || v ||2 = , then the space is called a Hilbert space.

Inner product can be thought of generalization of scalar product of vectors.

1) start with a finite dimensional vector space V(n) over the real- or complex numbers, note, this is a n-tupel space.

2) the dual space is isomorphic to V(n), and it is a therefore a second n-tupel space.

3) the bilinear form (x,y) maps to the field R or C, and may be written as "dot-product" or "scalar" product. (depends on the bases) 

4) The orthogonal- or unitary group keeps the bilinear productinvariant

A vector space (linear space) X over a field F (ℝ or ℂ) is called an inner product space if there is a function 〈⋅.⋅〉:X×X→F such that

(1) 〈x,x〉 ≥ 0

(1a) 〈x,x〉 = 0 if and only if x = 0

(2) 〈x+y,z〉 = 〈x,z〉 + 〈y,z〉

(3) 〈cx,y〉 = c〈x,y〉 for all scalars c ∈ F

(4) 〈x,y〉 = conjugate{〈y,x〉}.

〈x,y〉 is call the inner product of x and y. For example dot product is an inner product.

A vector space X together with an inner product from X×X to ℂ (respectively, ℝ) is referred to as a unitary (respectively, Euclidean) space.

A Banach space is a complete normed linear space (complete with respect to the induce metric).

Thus, a normed space X is a Banach space if every Cauchy sequence in

X converges.

A Hilbert space is a Banach space whose norm arises form an inner product.

Hilbert and Euclidean spaces are just particular cases of inner product spaces. A Hilbert space is an inner product space which happens to be a Banach space (with norm coming from the  inner product) and a Euclidean space  is the case X=Rn , with inner product = the dot product.

very smart, a nontrivial Hilbert/Banach space is first of all infinite dimensional and complete, Otherwise it is just Rn or Cn.

There was also the second part of the question,

"How and why is the concept of inner product spaces developed?"

As a possible answer to this question the very beginning of the §10 of Loomis' book "An introduction to abstract harmonic analysis" may be quoted:

"A Hilbert space H is a Banach space in which the norm satisfies an extra requirement which permits the introduction of the notion of perpendicularity. Perhaps the neatest formulation of this extra property is the parallelogram law:

llx+y||^2 + llx-y||^2 = 2[||x||^2 + ||y||^2]."

This permits also the introduction of inner product as is well known (and shown in the cited book and, of course, many other books).

Inner product is a generalization of a dot product or scalar product defined in 2-space.

It allows to define norm ,orthogonality and bilinear form  in n-space (as dot product for lenght , perpendicularity and area in 2-space) . Dot product measure an area like cross product, but the value is different.

Outer product is not related to inner product . The result of inner product is a scalar whereas the result of outer product is a matrix. The trace of this matrix is the inner product.

Exterior product is a generalization of a cross product defined on 2-space in order to measure area , volume ..... This concept completes the inner product as cross product completes dot product in 2-space.