I'm currently trying to solve the following problem:
$<−, −> : C^4 × C^4 → C$
Can we say that $<x, y> = x_1*y_1 + x_2*y_2 + x_3*y_3 + x_4*$ say` it's an inner product?
My approach is to show that the function satisfies the following properties:
- Linearity in the first vector: $ax+by=(ax_1+by_1,....,ax_n+by_n) <ax+by, x> = (ax_1+by_1)*z_1 + ... + (ax_n+by_n)*z_n = a<x,z> + b<y,z>$
- Conjugate symmetry: $<x,y>=<y,x> <y,x> = y_1*x_1 + y_2*x_2 + y_3*x_3 + y_4*x_4 = x_1*y_1 + x_2*y_2 + x_3*y_3 + x_4*y_4 = <x,y>$
- Non-negativity: $<x,x> = ||x_1||^2 + ||x_2||^2 + ||x_3||^2 + ||x_4||^2$
- Non-degeneracy: $<x,x> \ \text{iff} \ x_1=x_2=x_3=x_4=0$
This implies that $<x,y>$ is indeed an inner product. However, the answer is opposite. Could you explain me where I made a mistake, please?
an (Hemitian) inner product on a complex vector $V$ space must be anti-linear wrt second slot or conjugate-symmetric, that is : $$ \forall x,y \in V, \langle x,y \rangle =\overline{\langle y,x \rangle}$$ this property ensures that $$ \forall x \in V, \langle x,x \rangle \in \mathbb{R} $$ which makes positivity of $\langle x,x \rangle$ meaninfull. if you take $x=y=(i,0,0,0)$, you get:$\langle x,x \rangle <0$, which contradicts the axiom of positivity, so the given bilinear function is not an inner product.