Understanding problems of space I've been trying to understand the concept of space for some time now, but I still can't grasp the essence of it. In high school math we've been using 2D- and 3D- Euclidean space. Now that I am visiting advance math courses I hear about Hilbert spaces and Banach spaces and so on... 
Currently my understanding is that a Hilbert space has infinitely many dimensions and is complete. Correct?
I kept reading about spaces and I read things I don't understand:


*

*There exist spaces without a distance measure. What are they good for?

*Spaces can contain different numbers, e.g. some allow complex numbers, some don't. What is that good for? Don't I restrict my computations if I have a space that e.g. only contains rational numbers?

*How is the inner product defined for a space with infinitely many dimensions?

*Are there simple examples of a function space and a sequence space that I can understand?

*How does the space influence my computations? Well I understand, that the knowledge about being in a certain space we can make some assumptions (at least that's what my teacher does) but can anyone give an example where a different space would lead to a different result?

 A: 
  
*
  
*There exist spaces without a distance measure. What are they good for?
  

Yes, there are things called topological spaces, which don't necessarily have a "distance" function. Even if you don't have a "distance" function, you can still describe the "separation" of points using things called "neighbourhoods". This is a pretty abstract idea, and it probably won't be much use to you unless you study more advanced areas of mathematics. You can get a lot done using "metric" spaces, which are ones that do have a distance function.

  
*
  
*Spaces can contain different numbers ... Don't I restrict my computations 
  

Yes, but sometimes you want to restrict things. Suppose you're doing something as simple as solving a quadratic equation. Depending on your application, sometimes you want complex roots, and sometimes you only want real ones. 

  
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*How is the inner product defined for a space with infinitely many dimensions?
  

You're probably familiar with the $n$-dimensional space, in which each object is an $n$-tuple of real numbers $(x_1, x_2, \ldots, x_n)$. There, the inner product is defined by $\left<x,y\right> = \sum_{i=1}^n x_iy_i$. Now let $n$ become infinite, so that you have infinite sequences $(x_1,x_2, \ldots)$. Then you can define an inner product by $\left<x,y\right> = \sum_{i=1}^\infty x_iy_i$. There are some details to worry about, but that's the basic idea.
Another example is spaces of functions. You can define an inner product of two functions $x(t)$ and $y(t)$ by an integral: $\left<x,y\right> = \int x(t)y(t)\,dt$.

  
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*Are there simple examples of a function space and a sequence space that I can understand?
  

See above.

  
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*How does the space influence my computations? 
  

Let's use functions, again, as an example. Specifically, real-valued functions defined on the interval $[0,1]$. There are (at least) two obvious ways to measure the "distance" between two functions:
$$
\text{Using the area between their graphs:  } \quad \text{dist}(x,y) = \int_0^1|x(t)-y(t)|dt
$$
$$
\text{Using their maximum deviation:  } \quad \text{dist}(x,y) = \max\{|x(t)-y(t)| : 0 \le t \le 1\}
$$
These two ways of measuring distance give you two different "spaces". The objects in the space are the same in either case, but the way we measure distance is different. And these two different ways of measuring distance give different results. Imagine two functions that are very close to each other except in a tiny narrow region, where they differ by a value of 100. If you measure distance using the area method, it will be very small, but measuring with the maximum deviation method will give you an answer of 100.
I should emphasize that the discussion above is very sloppy and informal. I was just trying to convey some concepts, and there are many details that you need to worry about to make all this rigorous. But, you are a student of bioinformatics, not mathematics, so maybe intuitive understanding is more important to you than rigor. That's for you to decide.
