[Math] inner product and hermitian scalar product

Question

suppose $\underline x,\underline y\in\mathbb C^{n\times 1}$ then because the two vectors are in complex vector field, the definition of their inner product will be:
$$\langle\underline x,\underline y\rangle=\sum_{i=1}^n x_iy_i^*\qquad\text{* represents conjugate}$$
then it seems that the definition for hermitian scalar product of these two vectors is the same
$$\langle\underline x|\underline y\rangle=\underline x^T.\underline y^*=\sum_{i=1}^n x_iy_i^*\qquad\text{T represents transpose}$$
Are inner product and hermitian scalar product the same concept in vector spaces defined on complex numbers field?

Answer 1

A scalar product is just another name for inner product. Let V be a complex vector space. A Hermitian inner product on V is any function $$\langle , \rangle : V × V \rightarrow \mathbb{C},$$ that satisfies these axioms:

1. $$\langle u, v \rangle = \langle v, u \rangle $$

2. $$\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle \text{ and}$$

   $$\langle u, v + w \rangle = \langle u, v \rangle + \langle u, w \rangle $$

3. $$\langle cu, v \rangle = \overline{c} \langle u, v \rangle \text{ and}$$

   $$\langle u, cv \rangle = c\langle u, v \rangle .$$

4. $\langle u, u \rangle $ is a non-negative real number and $\langle u, u \rangle = 0$ if and only if $\langle u = 0 \rangle $.

What you've defined is just an example of a Hermitian inner product.