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Why is Fourier transforms of real-valued functions are symmetrical around the 0Hz

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pankaj jha

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Hello all !!!

Can anyone explain me

From theory, Fourier transforms is a even function only when the input signal is both real-valued and even function in time domain.
Then
1) why The Fourier transforms of real-valued functions are symmetrical around the 0 Hz axis.
2) in all the figures explaining sampling of real signals, the frequency domain representation of the input time domain signal is always symmetric about 0 Hz (DC).
3) is the input signal implicitly assumed to be real and even??
4) is practical real world cases the input signal implicitly assumed to be real and even??

Explanation with referred pdfs/books/web links are welcome.
 

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