Complex Inner-Product Spaces: True or False?

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Homework Statement



In an arbitrary complex inner-product space V which of the following is not always true?

a. <αu + βv, w> = α<u, w> + β<v, w>
b. |<u, v>|2 ≤ <u, u> <v, w>
c. <u, αv> = α<u, v>
d. <0, u> = 0

Homework Equations


None


The Attempt at a Solution



The correct answer must be a, but I don't know what to do with b.

a. False. If the constants are complex, they will be conjugated.
b. Any ideas?
c. True. Constants in the second argument will not be conjugated.
d. True. Multiplying a complex number by zero still results in a zero.
 
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If b) were |<u,v>|^2<=<u,u>*<v,v> it would be the Cauchy-Schwarz inequality. It's true. Is the 'w' a typo?
 
Dick said:
If b) were |<u,v>|^2<=<u,u>*<v,v> it would be the Cauchy-Schwarz inequality. It's true. Is the 'w' a typo?

Well, my professor has a 'w' on the assignment, but you must be right.

Thanks again!
 
Which argument gets the conjugate depends on whether it is a physics text or a math text. So consult the text.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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