differences between hilbert and euclidian spaces

[bhupala]

hi i have adoubt like what is the difference between hilbert space and euclidean space.

and also i have doubt like is the subspace same as subset?

thnx

regards

sri hari

 

[steve10]

euclidean space --- finite dimensional
hilbert space ---- infinite dimensional

subset --- a pure set without topology and algeraic requiement. For example, in 3-D, all points like {n,0,0} (n>0) consist of a subset.

subspace ---- there are some topology or algebraic requirement. In a Hilbert space, the most important algebraic requirement is that the subspace must be closed with respect to addition and constant multiplication. For example, all points like {n,0,0} do not consist of subspace, because -1 * n will no longer belong to this set (not closed by constant multipplication).

 

[bhupala]

i m sorry to disturb u again and i am basically electrical engineer and taking a course in mathematical methods and algorithms for signal processing which involves lot of stuff like this which i m very much less aware of as i have not taken any courses on functioanal analysis. So can u feel free to explain what do u mean by toplogy first?

thank u

sri hari

 

[steve10]

It's my fault. Forgive me for that bad term.
For a space or a subspace, not only do you have an algebraic structure (explained in the previous post like addition, constant multiplication,...), but you also have another structure which mainly measure the relationship between any two points. For example, in a Hilbert space, if x=(x1,x2,x3, ...), y=(y1,y2,y3,...), you may define the distance:
dist(x,y)=((x1-y1)^2 + (x2-y2)^2 + ...)^0.5.
In some more complicated spaces, you may not be able to define the distance between any two points like that. You may define a "norm", which essentially acts like the distance in simple cases. All those things, like distances, norms, even some more complicated forms, ... are all called topology structure.
A subspace of a hilbert space needs at least one topology structure which is usually inherited from the space itself.

 

[bhupala]

in euclidian space also u use the same formula for distance right?

so is euclidian space subset of Hilbert space?

sory if the queston is trivial.

thank u
srihari

 

[steve10]

No, I don't think it's trivial by any means. It takes time for you to get used to it.
Back to your question, "Euclidean space is a subset of Hilbert space". Well, It's like saying that tires are part of transportation. I can hardly say it is not right, but I can't agree with it either. I should say that the expression is not accurate. You should talk about specific tires and vehicles, something like "this tire is part of this car".
There are two very famous Hilbert spaces. One is the space of all squarely summable number sequences, which we talked in my previous post, usually denoted by l₂. The other is the space of all squarely integrable functions which has the following "distance":

dist(f,g)=(∫_{-∞}^{∞}|f(x)-g(x)|²dx)^{(1/2)}.

The corresponding space is denoted by L₂.
Now, it seems ok for you to say that the 3-D Euclidean space is a subset of l₂, but, apparently, it's NOT ok to say it is a subset of L₂.

 

[irfan1]

Hilbert space is obtained from an abstraction of the euclidean space. A more general term is the vector spaces. We call the infinite dimensional vector spaces as Hilbert space. For a brief discussion, you may check the introductory chapter of just any quantum mechanics book. For instance the first chapter of the book by R. Shankar (Principles of Quantum Mechanics) is an execellent introduction to the concepts of vector spaces.