inner product on vector space V

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SOLVED inner product on vector space V

True or false, and provide justification.

Let V be the vector space of continuous functions on [-Pi/2, Pi/2]. Define a pairing < - , - > on V by the formula

Then, < - , - > is an inner product on V .

I have no idea how to approach this, or even what it is asking... any help?
 
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Opalg

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True or false, and provide justification.

Let V be the vector space of continuous functions on [-Pi/2, Pi/2]. Define a pairing < - , - > on V by the formula

Then, < - , - > is an inner product on V .

I have no idea how to approach this, or even what it is asking... any help?
You just have to check that the inner product axioms are satisfied. So, is it true that

• \(\displaystyle \langle f,g+h\rangle = \langle f,g\rangle + \langle f,h\rangle,\)
• \(\displaystyle \langle g,f\rangle = \langle f,g\rangle ,\)
• \(\displaystyle \langle \lambda f,g\rangle = \lambda\langle f,g\rangle ,\)
• \(\displaystyle \langle f,f\rangle \geqslant0 ,\) with equality only if \(\displaystyle f=0\) ?

If all of those properties hold then that formula defines an inner product on V. If any one of them fails then it does not.
 
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