I'm trying to make an alternative (but equivalent) definition of an inner product. I prefer to use arbitrary sums with $\sum_i v_i$ instead of sum of just two vectors $v+w$, and few but clear axioms.
An inner product is a function $ \langle \cdot , \cdot \rangle : V \times V \to \mathbb{K}$ with $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, that satisfies
This is my list of axioms, I'll call it a $B$-inner product, I need help to know if this is correct:
B1. $\langle \sum_i \lambda_i v_i , \sum_j \mu_j w_j \rangle = \sum_i \sum_j \lambda_i \overline{\mu_j} \langle v_i, w_j \rangle $
for arbitrary $v_i, w_j \in V$, $\lambda_i, \mu_j \in \mathbb{K}$
B2. $ \langle v,v \rangle = 0 \Rightarrow v=0$.
I know the actual definition of inner product (I call it a $A$-inner product) is
A1. $ \langle v_1 + v_2, w \rangle = \langle v_1,w \rangle + \langle v_2, w \rangle $
A2. $ \langle \lambda v, w \rangle = \lambda \langle v,w \rangle $
A3. $ \langle v,w \rangle = \overline{ \langle w, v \rangle }$
A4.1 $ \langle v,v \rangle \geq 0$
A4.2 $\langle v,v \rangle = 0 \Leftrightarrow v=0$
Clearly an $A$-inner product is a $B$-inner product.
And clearly $B1$ implies $A1$ and $A2$. At least in finite dimension I think it also implies $A3$ (but not sure): I write $v$ and $w$ in an orthonormal basis, apply $B1$ on $\langle w,v \rangle$, and conjugate, it gives the same than the product $\langle v,w \rangle$.
I think it also satisfy $A4.1$ as the product of a complex number by its conjugate always gives a non-negative real number.
Is my definition equivalent? Do I also need $v=0 \Rightarrow \langle v,v \rangle = 0$? What other axioms should I add to make it equivalent?
Thanks.
Assuming the sums $\sum_i,\sum_j$ are over finitely many indices, your definition is equivalent to the standard one.