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

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
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: $$\begin{aligned}T(\lambda u+\mu v)&=<\lambda u+\mu v,y>z\\&=(\lambda<u,y>+\mu<v,y>)z\\&=\lambda<u,y>z+\mu<v,y>z\\&=\lambda T(u)+\mu T(v)\\&\Rightarrow T\mbox{ is linear}\end{aligned}$$