What is the hermitian space? What is the hermitian space?
 What is the simple example for it? Is $\mathbb{C}^3$ example for hermitian space?
 A: A space consists not only of the underlying set, but what additional structure defined on it. The Euclidean space $\mathbb{R}^n$ for example can be equipped with the scalar product $\mathbb{x} \cdot \mathbb{y} = \sum_{i=1}^n x_i y_i$  as an inner product that gives it "length" and "orthogonality".
But for complex spaces, the scalar product is not suitable. Take $\mathbb{C}^2$ for example, we can find $\mathbb{x} = (i, 1) \neq 0$ with $\mathbb{x} \cdot \mathbb{x} = 0$ violating positive-definiteness of inner products.
The Hermitian product defined by $\mathbb{x} \cdot \mathbb{y} = \sum_{i=1}^n \overline x_i y_i$ makes $\mathbb{C}^n$ an inner product space, thus with that data $\mathbb{C}^n$ is a Hermitian space.
We can now proceed to define the norm as $\|\mathbb{x}\| = \sqrt{\mathbb{x}\cdot \mathbb{x}}$, the distance between two points $\mathbb{a}, \mathbb{b}$ by $d(\mathbb{a},\mathbb{b}) = \| \mathbb{b} - \mathbb{a}\|$, orthogonality by $\mathbb{x} \cdot \mathbb{y} = 0$. Analogues results concerning orthonormal basis, triangle inequality, Cauchy-Schwarz inequality, Pythagorean theorem also holds. 
See This lecture note, by W. Allard.
