MHB Gregory's question at Yahoo Answers (Inner product space)

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To prove that the transformation T defined by T(x) = <x,y>z is linear, one must demonstrate that it satisfies the properties of linearity for all vectors u, v in the vector space V and scalars λ, μ in the field F (either ℝ or ℂ). By applying the definition of the inner product and the linearity of the inner product itself, it can be shown that T(λu + μv) equals λT(u) + μT(v). This confirms that T adheres to the linearity conditions, thus proving that T is indeed a linear transformation. The discussion emphasizes the importance of using the axioms of inner products to establish linearity in vector spaces. The conclusion affirms the successful proof of T's linearity.
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Hello Gregory,

If $V$ is the given vector space over the field $\mathbb{F}\;(\mathbb{R}$ or $\mathbb{C}$), for all $\lambda,\mu\in\mathbb{F}$ scalars, for all $u,v\in V$ vectors and using the axioms of inner product: $$\begin{aligned}T(\lambda u+\mu v)&=<\lambda u+\mu v,y>z\\&=(\lambda<u,y>+\mu<v,y>)z\\&=\lambda<u,y>z+\mu<v,y>z\\&=\lambda T(u)+\mu T(v)\\&\Rightarrow T\mbox{ is linear}\end{aligned}$$
 
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