Difference between hilbert space,vector space and manifold?

[aditya23456]

Difference between hilbert space,vector space and manifold??

Physically what do they mean? I m really confused imagining them..Explanation with example would help me to understand there application ..THanks in advance

 

[Office_Shredder]

A hilbert space is simply a vector space with an inner product. Manifolds and vector spaces have very little to do with each other... Do you know what the definitions of these are?

An example of a manifold is a sphere of radius one, or a graph of a function

 

[Matterwave]

These three concepts are very precisely defined concepts in mathematics. All three are topological spaces endowed with some extra properties.

A vector space is, roughly speaking, a space in which the addition and subtraction of points (called vectors) in the space is possible, along with a few other properties like being able to multiply points with scalars and this multiplication being distributive. I don't remember all of the exact properties a vector space must have, but you can look it up on wikipedia.

A Hilbert space is a vector space with a defined inner product. This means that in addition to all the properties of a vector space, I can additionally take any two vectors and assign to them a positive-definite real number. This assignment has to satisfy some additional properties. It has to be 0 only if one of the vectors I give it is 0. And it has to satisfy the triangle inequality (you can look this up).

A manifold is a topological space which is locally Euclidean. This means that it can be covered by an atlas of charts (mapping of points on the manifold to R^n). We call such an object a topological manifold. If, in addition, the atlas's charts are compatible with each other in that their coordinates in the intersections are differentiable, smooth, functions of each other, then we call them differentiable manifolds. Manifolds do not need to be vector spaces.

If I'm imprecise somewhere, I hope a mathematician comes and corrects me hehe.

 

[homeomorphic]

No, this mathematician is going to come along and be even more imprecise. Mathematicians aren't always completely precise if the situation doesn't call for it.

I'm not sure if these things exactly have a physical meaning. The physical meaning is dependent on what particular example you are looking at.

A manifold is basically just a higher-dimensional surface. Locally, you can put coordinates on it. Or you could say, "it looks like R^n if you zoom in on it". So, the best example is a surface in 3 dimensional space. Another example is 3 dimensional space itself. Another example would be a configuration space for a physical system. This is a space whose points correspond to configurations of the system. For example, a robot arm, consisting of a rigid upper arm and lower arm would have a torus as its configuration space, since you need two angles to specify the position of the robot arm or a point of the torus. Another example is R^3n, which is the configuration space of n point particles (allowing them to occupy the same point). There are some other important physical examples, like velocity space (tangent bundle of configuration space), phase space (cotangent bundle of configuration space), and space-time.

Vector space. Just means a place where you can add and multiply by scalars. These can have many different interpretations. It's not true that vector spaces and manifolds have nothing to do with each other. R^n is perhaps the most important vector space, but it's also a manifold. Not only that, but it's the space that real manifolds are built out of. The canonical example of a real vector space is just ordinary n-dimensional space, with some point that you choose as the origin. But you think of each point as an arrow. And you can add arrows tail to tip. So, that's the main example, but it's far from the only one. Other examples might have a different physical meaning. For a complex vector-space, the generic physical example I would have in mind is a wave function on a space consisting of only finitely many points. That's isomorphic to C^n.

And you can also have more general vector-spaces over different "fields", as the mathematical jargon goes. I would imagine some of them might have a physical interpretation, but they are fairly algebraic in nature.

A Hilbert space is a vector space with a sort of "dot product". This does have a physical meaning, but it's hard to convey in one paragraph, so I will be very vague. In quantum mechanics, you could think of it as having something to do with probability amplitudes, which are complex numbers whose squared modulus is proportional to the probability (of being measured to be in that state, which will be some value associated to an observable). For other waves, the analogous thing would be energy density, rather than probability density.

 

[HallsofIvy]

A hilbert space is simply a vector space with an inner product.

That's not quite correct. A vector space with an inner product is an "inner product space". If that inner product space is "complete" (Cauchy sequences converge) then it is a "Hilbert Space". Of course, in general vector spaces we do not have a notion of "convergence" of an infinite sequences of vectors. As soon as we have an inner product, we can then define a "norm" and so a "metric" on the space and have limits.

Given an inner product, <u, v>, we can define a norm by $|v|=\sqrt{<v, v>}$ but there exist norms not derivable from an inner product. A vector space with a norm (a "normed space") where Cauchy sequences converge is called a "Banach space"/

And given a norm, |v|, we can define a metric, d(u, v)= |u- v|, but there exist metrics not derivable from a norm. A vector space with a metric where Cauchy sequences converge is called a "Frechet Space".

 

[Fredrik]

These three concepts are very precisely defined concepts in mathematics. All three are topological spaces endowed with some extra properties.

To define a vector space, you just need the underlying set, a field of scalars, the scalar multiplication operation and the addition operation. You don't need a topology.

A Hilbert space is a vector space with a defined inner product. This means that in addition to all the properties of a vector space, I can additionally take any two vectors and assign to them a positive-definite real number. This assignment has to satisfy some additional properties. It has to be 0 only if one of the vectors I give it is 0. And it has to satisfy the triangle inequality (you can look this up).

What you're describing is an inner product space. The inner product takes a pair of vectors to a scalar. If the field of scalars is ℂ, it takes a pair of vectors to a complex number. However, for all x, <x,x> is real.

A Hilbert space is a complete inner product space, i.e. an inner product space in which every Cauchy sequence is convergent.

 

[Matterwave]

If you have an inner product, then you have a norm. If you have a norm, then you have a metric. If you have a metric then you have a metric topology. So I don't believe I'm wrong in saying that a Hilbert space is a topological space with inner product.

I forgot the additional condition that the inner product be complete. Sorry about that.EDIT: Sorry, I remembered what I posted wrong. I must have been in a rush today morning and made the mistake of claiming a vector space is a topological space. I was thinking about a Banach space. Sorry.

 

[homeomorphic]

By the way, the completeness condition only matters if the vector space is infinite-dimensional. Finite-dimensional ones are all complete automatically.

 

[homeomorphic]

Also, at least one motivation for the completeness condition is that it allows you to have a good theory of projections.

And any vector space with an inner product has a completion, so it's sitting inside a Hilbert space, possibly as a proper subspace.

 

[Matterwave]

That's good to know. =]