What is hilbert space and how can I visualize it?

[Mr. Peterman]

Hi. I'm trying to make sense of what hilbert space is. I have a book about it but it treats everything symbolicly and I'm just wondering if someone here can give me concrete examples of what an element of hilbert space can be and maybe show me an example of the properties of hilbert space actually working using actual numbers or functions or whatever. My current understanding or lack thereof is that an element of hilbert space is something like a linear combination of different functions with the coefficients of each function being the coordinates or something kind of like in linear algebra. Maybe an element is some complicated function of x and y with the x and y being the coordinates of hilbert space. I really don't have a clue. I'm also lacking a proper background to understand this stuff because most of the math I know was learned from mathematical physics books rather than math books and I haven't learned much past linear algebra like topology, differential geometry, or field theory although I've learned a little group theory. Most of what I learned about linear algebra is from mathematical physics books too. I would really appreciate it if someone could give me a good example with actual functions or numbers or tell me what a good book is to learn from that will give a person an intuitive understanding. I haven't seen any books that look promising in this respect. I'm afraid that most of them will drown me in a sea of symbolism that I won't understand without ever giving me a hint of actual functions or numbers that can bring it more down to earth for me. Thank you for your help. Maybe little by little I'll learn something here.

[Essay]

Hi. I'm trying to make sense of what hilbert space is. ...that will give a person an intuitive understanding.

 

http://mathnet.preprints.org/EMIS/journals/EJDE/Monographs/01/chpt1.pdf

It is addressed to beginning graduate students of mathematics, engineering, and the physical sciences. Thus, we have attempted to present it while presupposing a minimal background: the reader is assumed to have some prior acquaintance with the concepts of "linear" and "continuous" and also to believe L^2 is complete.

A problem is called well-posed if for each set of data there exists exactly one solution and this dependence of the solution on the data is continuous. To make this precise we must indicate the space from which the solution is obtained, the space from which the data may come, and the corresponding notion of continuity. Our goal in this book is to show that various types of problems are well-posed. These include boundary value problems for (stationary) elliptic partial differential equations and initial-boundary value problems for (time-dependent) equations of parabolic, hyperbolic, and pseudo-parabolic types. Also, we consider some nonlinear elliptic boundary value problems, variational or uni-lateral problems, and some methods of numerical approximation of solutions.

...

We recall that the (continuous) dual of a seminormed space is a Banach space. We shall show there is a natural correspondence between a Hilbert space H and its dual H'.

Holy Frijole!!!

 

As a fan of biophysics, perhaps I can offer this "image" that should be a bit more intuituve, if not too oversimplified:

 

Suppose you had an equation which described how the metabolism works when stressed by cold.... &

you had another equation which described how the metabolism works when stressed by heat....

 

If (using the two equations) you graphed the minimum limit of viability for cold conditions, and the maximum limit of viability for hot conditions, then the space between those two "lines," or limits, would be the Hilbert space (of viability based on temperature).

 

...if I get what they are saying above. Hopefully somebody can expand upon or correct my attempt at an intuitive description.

[Boof-head]

You can find a Hilbert space in any closed (complete) system which "uses" vectors.

Vectors and vector products (a product space) are easy to find in a system with fixed vectors, like 0 and 1 in electronic digital computers.

 

You need to show that a 0 and a 1 are both like rows (or columns) in a 2x2 matrix - 0 = (1,0), 1 = (0,1); and you're away laughing.

 

If you can induce or deduce the identity 2x2 matrix I, you have a 'complete' space; then you need to find operators that invert these vectors (so a 0 becomes a 1, and a 1 becomes a 0), which is the 'communication' part - a signal is ultimately a change (or a series of changes) in space and time.

 

Then if you can find operators that are equivalent to a logic - like Boolean logic, AND and OR - you have a computational space, with a computational "basis". The basis is fixed in classical computation so the result is always a transform of the input or start condition.

The simplest classical digital basis is binary logic, so the symmetries in space and time imply an unitary symmetry group and Lie algebras (according to Seth Lloyd - but I'm still trying to get my head around this).

 

In quantum circuits, the basis can be swapped around (like a Boolean operator), and the input must not be 'destroyed' or erased, since the output is the input - QIC must be reversible. This means (sort of) that you need to keep the original "image", and reflect it in a sort of quantum mirror. At least that's an analogy I use.

 

Try googling stuff about computation and logic along with "hermitian operator" and "hilbert". You might try "orthogonal", "orthonormal" and "basis" too; there are some good freshman lectures on this stuff.

 

I could post a ref. to a good PDF about doing all the above (with algebra), which also explains Dirac notation (it's easy stuff, really) and all the other terminology.

However, I'm currently a member of the disenfranchised... a techno-snafu has appeared on the horizon, and 'home' is disconnected.

[Mr. Peterman]

I've read that an element of hilbert space can be a function. Is that what functional analysis is mostly about. Is there any use of this fact. I'm wondering if the components of these elements are the individual terms in an expansion of a function in an infinite series. Is that true or am I missing it. And if is true is that what all the stuff regarding infinite dimensions are about. I'm kind of lost.

[Boof-head]

Check out the start of this course; it's a good intro to the connection between classical and quantum probability, and tensor products

 

COMSM0214: Quantum Computation

[lemit]

I'm not good at math, but it seems that math isn't always required in theoretical physics.

 

Having said that, couldn't Hilbert Space be used to define the knowable universe, with Banach Space constituting the unknowable?

 

Thinking about something like that is much better than thinking about how late I am in dealing with my taxes.

 

--lemit

[maddog]

I've read that an element of hilbert space can be a function. Is that what functional analysis is mostly about. Is there any use of this fact. I'm wondering if the components of these elements are the individual terms in an expansion of a function in an infinite series. Is that true or am I missing it. And if is true is that what all the stuff regarding infinite dimensions are about. I'm kind of lost.

From my dealings in QM and some other Graduate Physics work, I learned that a Hilbert Space is simply an Infinite Dimensional Vector Space. This would mean you would have an infinite quantity of coordinates in your basis: xi where i -> infinity.

 

Try these links:

 

Hilbert space - Wikipedia, the free encyclopedia

Hilbert Space -- from Wolfram MathWorld

Hilbert space: Definition from Answers.com

 

Banach space is separate (though maybe related) is based in Complex algebra

 

where a function is decomposed

 

f(z) = u(z) + iv(z)

 

I think Functional Analysis derives from Banach space (as well as Spectral Analysis).

 

From group theory I think the group SL(R, n) is isomorphic to Hilbert Space as n -> infinity.

 

maddog

[Boof-head]

Decomposition definitely comes into vector spaces; not all Hilbert spaces are separable, all complex spaces with an inner product and a norm are: I think that might be restricted to even-numbered dimensions, because you can prepare bipartite and tripartite, for example, states - qbits and trits - in C^2 which will always be separable. Banach spaces are measurable, or have a 'complete algebra' in them that includes measurement operators/generators; FA leads to Maxwell and linear analysis of networks, EE and so on.

[Essay]

I've read that an element of hilbert space can be a function. Is that what functional analysis is mostly about. Is there any use of this fact. I'm wondering if the components of these elements are the individual terms in an expansion of a function in an infinite series. Is that true or am I missing it. And if is true is that what all the stuff regarding infinite dimensions are about. I'm kind of lost.

 

Well, I'm learning more and more, I think--that someday may make sense.

Thanks Boof-head... and DD too.

 

...and maybe I'm off, describing something other than Hilbert space, but I'll stick with my example above--and avoid the theoretical physics.

 

First I'll suggest that Hilbert space can (theoretically) be of infinite dimension, but it doesn't have to be; it can be constructed from whatever dimensions one is working with.

 

To explain the oversimplification above, in my first post... I put quotes around the word "line" because it isn't really a line, but a limit of space described by some multidimensional equation (describing metabolism/temperature in this case).

In my example above, the "line" plotting the equation of metabolism (stressed by hot temperatures) would be composed of many separate equations--each describing how a particular enzyme operated thru a temperature range, and how friction and viscosity of various metabolic systems change over a temperature range, and how regulatory mechanism work to compensate for temperature change and how dynamic and robust they are, etc.--all these equations combining to (create a "line" to) describe the limit of viable metabolism....

 

All these equations combining to (create a "line" to) describe the limit of viable metabolism....

Each of these equations (or functions) is a different dimension.

All of these equations are different dimensions (or elements)... defining the Hilbert space.

 

Well, it's defining one side of the Hilbert space; the other "line" setting the limit for "cold viability" would define the other side of the Hilbert space.

===

 

One could monitor hundreds of enzymes-along with dozens of systems and a few mechanisms--to create a Hilbert space (describing metabolism in this case) composed of several hundred dimensions.

 

Does this ring a more intuitive bell?