How do I prove that a bilinear map is symmetric in the given condition?

Question

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Let $X$ be a Banach space over $\mathbb{R}$ and $\lambda_1,\ldots,\lambda_n$ be linearly independent elements in $X^*$.

Define $X^+:=\{x\in X: \lambda_1(x)\geq 0,\ldots,\lambda_n(x)\geq 0\}$.

Let $T:X\times X \rightarrow \mathbb{R}$ be a continuous bilinear map such that $T(x,y)=T(y,x)$ for all $x,y\in X^+$.

How do I prove that $T(x,y)=T(y,x)$ for all $x,y\in X$?

Answer 1

Hints:

1. Go over to the quotient space of $X$ by factoring out the closed subspace $X^+ \cap (-X^+)$. Then, the cone $X^+$ after factoring is a proper closed affine cone.

2. In quotient space, which is a Riesz space (vector lattice), each element admits a decomposition into difference of positive elements. Transporting the decomposition back into $X$ and exploiting the bilinearity yield the statement.