Consider a class of inner product spaces
$$\langle \cdot,\cdot\rangle_{{\lambda}\in \Lambda}: R^n\times R^n\to R$$
parameterized by $\lambda \in \Lambda=\Delta(\{w_1,....,w_n\})$, the set of all probability distributions on the $n$ dimensional finite support.
Now suppose the class of norms $\|x\|_{\lambda}=\sqrt{\langle x,x \rangle_{\lambda}}$ satisfies the following linearity in probability assumption: for all $x\in R^n$ and all $\alpha\in [0,1]$, it holds
$$\alpha\langle x,x \rangle_{\lambda_1}+(1-\alpha)\langle x,x \rangle_{\lambda_2}=\langle x,x \rangle_{\alpha\lambda_1+(1-\alpha)\lambda_2}, \phantom{000} \forall \lambda_1, \lambda_2\in \Lambda$$
What additional assumptions are needed in order to conclude that the inner product space with parameter $\lambda$ must have the form of $L_2(\lambda)$, i.e.
$$\langle x,y\rangle_{\lambda}=\sum_{i=1}^n\lambda_i x_i y_i$$ ?
Would Riesz Theorem or other existing characterization of $L_2(\lambda)$ be useful? Thanks.
Suppose in $R^2$, for all $\lambda=(\lambda_1, \lambda_2)$, $\lambda_1+\lambda_2=1$ and $\lambda_1,\lambda_2\geq 0$, we have
$$\langle( 1-\lambda_1, \lambda_1),(1,-1)\rangle_{\lambda}=0$$
Can we conclude the functional form of $\langle.,.\rangle_{\lambda\in \Lambda}$?