Why do we want an inner Product to satisfy the following criteria?

Question

When trying to understand euclidean Vectorspace, I asked myself the question, why one would define the inner Product by the following three criteria:

1) linearity in the first argument

2) Symmetry and

3) positive definiteness

Why have mathematicians invented the inner product? What special properties does such a function have? It must have had a reason to be invented. So far I couldn't really find a proper answer to this question on the internet, since it is frankly an unusual question to ask. Thank you very much for your answer in advance!

Answer 1

Why need linearity in the first argument:

$$\|\lambda u\|^2=⟨\lambda u,\lambda u⟩=\lambda\overline{\lambda}⟨u,u⟩$$ $\lambda\overline{\lambda}$ is always a real number because of linearity in the first argument you get compatibility with the definition of a norm. As we need inner product to define norm also. As @celtschk mentioned linearity can be define on second argument also$(\text{similar reasoning})$.

Why need symmetry and positive definiteness:

Symmetry is generally part of the definition. In linear algebra Inner product comes first axiomatically anything that has three property:
$(i)$ symmetric
$(ii)$ Distributive
$(iii)$ positive definiteness
personally I like to describe inner product as a definite positive symmetric bilinear form.

Why have mathematicians invented the inner product:

If you familiar with dot product then you can remember that it was used to find angle and norm. But in abstract you can't define norm and angle so easily. Think about polynomials. How could you define length,angle of polynomials$?$ So, In abstract sense inner product come first(axiomatically) to rescue them.

Answer 2

The concept of inner product (or dot product) is not evident in itself.

A very plausible explanation is that it is a quantity that appears in some computations.

In particular in the computation of the norm of the sum of two vectors ; let us detail it in $\mathbb{R^2}$ :

Assume that we have two vectors $\vec{V_k}=(x_k,y_k)$, $k=1,2$.

The squared norm of their sum is :

$$(x_1+x_2)^2+(y_1+y_2)^2=(x_1^2+y_1^2)+(x_2^2+y_2^2)+2(x_1y_1+x_2y_2)\tag{1}$$

i.e., the sum of the squared norm of $\vec{V_1}$ and the squared norm of $\vec{V_2}$ plus a certain quantity (of course, this is the dot product).

Moreover, one can write (1) under the form

$$\|\vec{V_1}+\vec{V_2}\|^2=\|\vec{V_1}\|^2+\|\vec{V_2}\|^2+2 \vec{V_1} . \vec{V_2}\tag{2}$$

in order that it looks completely analogous to the classical algebraic formula

$$(a+b)^2=a^2+b^2+2ab$$

In this way, we have at the same time the concept and an appropriate notation !

Of course, a particular case of (2) is when $\vec{V_1} \perp \vec{V_2}$ : where one can apply Pythagoras theorem : $\|\vec{V_1}+\vec{V_2}\|^2=\|\vec{V_1}\|^2+\|\vec{V_2}\|^2$ which means that this case appears if and only if the inner product is zero, providing a criteria of orthogonality (in fact already known before), providing a very handy way to include the inner product into analytic geometry computations.

Historical note: Inner product appeared, in a shy manner, in the second half of the 19th century. The physicist Gibbs played an eminent rôle in the officialization at the very end of the 19th century, at the same time of the inner product and of the cross product, with harsh discussions in both cases about these "monsters":

• the inner product is said a product of two vectors but is a number, not a vector

• the cross product, while being a vector, is plagued by its anti-commutativity ($\vec{V_2} \times \vec{V_1}=-\vec{V_1} \times \vec{V_2}$).