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daniel_i_l

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1) C[-1,2] is a space of all continues functions f: [-1,2] -> C (complex)

Is:

[tex] <f,g> = \int_{-1}^{2}|f(t) + g(t)|dt [/tex]

an inner product of C[-1,2]?

I think that the answer is no because:

[tex] <f+g, h> \neq <f,h> + <g,h>[/tex]

for some f and g. this can happen when all the functions are positive and so:

|f(t) + h(t) + g(t)| doesn't equal |f(t) + h(t)| + |g(t) + h(t)|

2)

V is a space of all real functions with defined double derivatives in the interval

[tex] [-\pi, \pi] [/tex]

we define:

[tex] <f,g> = f(-\pi)g(-\pi) + \int_{-\pi}^{\pi}f''(x)g''(x)dx [/tex]

is <f,g> an inner product of V?

also here i think that the answer is no because <f,f> can equal zero even if all of f isn't 0, this can happen if f(-pi) is zero and the double derivative is 0 everywere (contiues slope).

am i correct?

Thanks.

Is:

[tex] <f,g> = \int_{-1}^{2}|f(t) + g(t)|dt [/tex]

an inner product of C[-1,2]?

I think that the answer is no because:

[tex] <f+g, h> \neq <f,h> + <g,h>[/tex]

for some f and g. this can happen when all the functions are positive and so:

|f(t) + h(t) + g(t)| doesn't equal |f(t) + h(t)| + |g(t) + h(t)|

2)

V is a space of all real functions with defined double derivatives in the interval

[tex] [-\pi, \pi] [/tex]

we define:

[tex] <f,g> = f(-\pi)g(-\pi) + \int_{-\pi}^{\pi}f''(x)g''(x)dx [/tex]

is <f,g> an inner product of V?

also here i think that the answer is no because <f,f> can equal zero even if all of f isn't 0, this can happen if f(-pi) is zero and the double derivative is 0 everywere (contiues slope).

am i correct?

Thanks.

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