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A Hilbert Space is a complete inner product space.

My first question: From the definition above, is it safe to say that every sequence in a Hilbert Space converges? And so can we say that Hilbert Spaces only contain Cauchy sequences?

Second question: These 'sequences' that we talk about in Hilbert spaces, can they be sequences of anything? Like sequences of functions, sequences of elements (like complex numbers), sequences of inner products?

Third question: Why is [itex]l_2[/itex], the set of all square-summable sequences, the only [itex]l_p[/itex] space that is a Hilbert space?

Fourth question: The Cauchy-Schwartz Inequality

[tex]|\langle x,y \rangle \leq \|x\|\|y\|[/tex]

This always holds in a Hilbert space right? In which instances does it not hold?

Fifth question: What are the differences between Hilbert Spaces and Banach Spaces? The only thing I know is that Hilbert spaces are 'nicer' than Banach spaces because of something to do with the parallelogram law - which in turn makes Fourier analysis work better or something...I dont really know.

That will do for now.

My first question: From the definition above, is it safe to say that every sequence in a Hilbert Space converges? And so can we say that Hilbert Spaces only contain Cauchy sequences?

Second question: These 'sequences' that we talk about in Hilbert spaces, can they be sequences of anything? Like sequences of functions, sequences of elements (like complex numbers), sequences of inner products?

Third question: Why is [itex]l_2[/itex], the set of all square-summable sequences, the only [itex]l_p[/itex] space that is a Hilbert space?

Fourth question: The Cauchy-Schwartz Inequality

[tex]|\langle x,y \rangle \leq \|x\|\|y\|[/tex]

This always holds in a Hilbert space right? In which instances does it not hold?

Fifth question: What are the differences between Hilbert Spaces and Banach Spaces? The only thing I know is that Hilbert spaces are 'nicer' than Banach spaces because of something to do with the parallelogram law - which in turn makes Fourier analysis work better or something...I dont really know.

That will do for now.

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