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I am taking a class in functional analysis, but it's been a very long time since I've studied this topic. I'm even struggling to remember the pre-requisites materials. I am trying to re-learn (and learn) as much as possible and I've been researching many topics, but failing to see a big picture.

One thing I struggle with is the terminologies, which makes it hard for me to even read the answers to the questions I try to ask - can someone please tell me whether there is a "layman's term" for the following ideas? How do they relate to each other?

  1. Is a Hilbert space the same thing as writing $"l^{2}"$? What does it mean to have $l^{2}(\mathbb{N})$ as opposed to $l^{2}(\mathbb{R})$?

  2. I don't know if I understand inner product space - my research tells me its essentially the same as a cross-product - is this the right intuition? Is the inner product space denoted like $(f,g)$ or $||f||||g||$? (Are those two notations equivalent?) Is it the same thing as if I insert a "dot" like $||f|| \cdot ||g||$?

  3. Is the "norm" the same as the "measure," or what we'd call the "length" in a Euclidean space? Do we call it a "norm" simply as a more generalized version of "length"/"distance" if we are not in Euclidean space? To me it seems like the same idea as absolute value $|f|$, so why do we use 2 bars like $||f||$ instead of one?

  4. If a function is complete, that means every Cauchy Sequence converges...ok, I'm guessing there is some conventional "technique" for showing this? I don't understand how one can show every Cauchy sequence converges (I need to refresh of Cauchy Sequences too...) but is there a way i can think of this idea that at least will seem like it's a plausible task? It seems like this technique will get used a lot in what I need to study...

...So, I would please appreciate some confirmation that I am thinking about these words in the right way. Thank you.

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$1)$ A Hilbert space is an inner product space (this induces a norm) that is complete with respect to the induced norm. Every seperable real Hilbert space is isomorphic to $l^2$ (the space of square summable real sequences) as a Hilbert space, i.e. there is a linear map $T: H \rightarrow l^2$ such that $T$ is continuous, linear, and preserves the inner product. It is not always good to think of Hilbert spaces as "just $l^2$. Those spaces are probably the same. You can also consider two sides sequences.

$2)$ It is highly not a cross product. It is literally a dot product. The first notation you use are standard and the others mean different things.

$3)$ Many people use one bars precisely because it doesn't matter. A norm is a special kind of metric (as in metric space). It gives you notions of distance. In Hilbert spaces you also have angles.

$4)$ A function cannot be complete. Spaces are complete. Notice this is in the definition of a Hilbert and Banach spaces. It is crucial you understand a Cauchy sequence before trying to learn functional analysis. A complete space is one that has "no holes" that means we can limits without worry about the limit being in the space. The rationals are not complete , because limits of them can be irrational. The reals are complete.

You really should look up a text on functional analysis, there's a lot you're missing.

Best of luck!

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  • $\begingroup$ Thank you - Yes my textbook is very helpful, if I know what I am looking for :) This answer helps me a LOT too - I have so much to learn. And I will surely take your advice to learn Cauchy Sequences again before continuing too far. It sounds important :p Thanks so much! $\endgroup$ – PBJ Jan 28 '18 at 2:46

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