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This problem is giving me trouble near the end of the question when it uses 〈Tv,w〉. I'm pretty sure I could solve it if it were 〈Tv,v〉using invertibility, but I am unsure what to do with this. any help is appreciated, thanks.

Let (V,〈·,·〉) be a real inner product space. We say that T is positive-definite if, for some r >0. 〈Tv,v〉≥r(||v||^2). Show that 〈〈v,w〉〉:=〈Tv,w〉 defines an inner product on V.

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Linearity and positive-defineteness follow immedietaly from the definition. Also, the usual definition of positive-definite operator also requires symmetry, which means that

$$ \langle Tv,w\rangle = \langle v,Tw\rangle $$

Symmetry of $\langle\langle,\rangle\rangle$ follows from this.

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