Understanding Hilbert Space: A Simplified Explanation

[ZaiKin786]

I was wondering what is Hilbert Space exactly?
I read the Wikipedia page, but its one of those situations u understand what your reading but don't full grasp the concept.
I was just hoping someone could explain it to me.

 

[Hurkyl]

Without knowing what you get and what you don't, we can't really do much better...

 

[arunma]

Without knowing what you get and what you don't, we can't really do much better...

This is true. Hilbert space is a pretty big topic (granted, most physicists just understand the parts that we need to get by).

I guess the best I can do is quote David Griffiths: Hilbert space is where wavefunctions live. Most wavefunctions you encounter are functions of one position variable. So the Hilbert space here is a function space. Some wavefunctions are two-component spinors, so these Hilbert spaces are vector spaces spanned by a 2x1 matrix. Does this make sense?

 

[DrGreg]

The motivation behind Hilbert spaces is to treat functions like vectors. You can add two functions together and multiply a function by a scalar. It is possible to define an "inner product" between two functions that behaves like the dot product of two vectors, which you can use to define the "length" (norm) of a function and the "angle" between two functions, and in particular defines what it means for two functions to be "orthogonal" to each other. You can decompose many functions as a (possibly infinite) sum of orthogonal basis functions. Differential equations can be treated similarly to matrix-vector equations.

A major application of Hilbert space theory is to the wavefunctions of quantum theory.

That's a simplified introduction; if you want more details you'll have to indicate "what you get and what you don't".

 

[HallsofIvy]

In addition to being a vector space with inner products (so we can define "length" and "angle"), a Hilbert space must also have the "Cauchy property": if {v_n} is a sequence of vectors in the space such that ||v_n- v_m|| goes to 0 as n and m to to infinity, independently, the {v_n} converges to some vector in the Hilbert space.

 

[crazygoofyman]

A Hilbert Space is the sum of the multiplicative inverses of all known vectors that are real-valued scalar functions.
R[2]\prod[/1/x]

 

[Hurkyl]

In addition to being a vector space with inner products (so we can define "length" and "angle"), a Hilbert space must also have the "Cauchy property": if {v_n} is a sequence of vectors in the space such that ||v_n- v_m|| goes to 0 as n and m to to infinity, independently, the {v_n} converges to some vector in the Hilbert space.

And because of this, calculus behaves relatively nicely.

 

[Fredrik]

I was wondering what is Hilbert Space exactly?
I read the Wikipedia page, but its one of those situations u understand what your reading but don't full grasp the concept.
I was just hoping someone could explain it to me.

If you already know that it's a complete inner product space, then...what Hurkyl said.

...which you can use to define the "length" (norm) of a function and the "angle" between two functions,

This only works for real Hilbert spaces. (I have claimed otherwise in this forum, so maybe I'm the one who gave you the wrong idea about this). The Cauchy-Schwartz inequality tells us that

$$\frac{|\langle x,y\rangle|}{\|x\|\|y\|}\leq 1$$

which together with the relationship $\vec x\cdot\vec y=|\vec x||\vec y|\cos\theta$ that holds for vectors in $\mathbb R^3$ suggests that we can define the angle by

$$\cos\theta=\frac{\langle x,y\rangle}{\|x\|\|y\|}$$

but this only makes sense if the numerator is real.

In addition to being a vector space with inner products (so we can define "length" and "angle"), a Hilbert space must also have the "Cauchy property": if {v_n} is a sequence of vectors in the space such that ||v_n- v_m|| goes to 0 as n and m to to infinity, independently[/color], the {v_n} converges to some vector in the Hilbert space.

The red part is misleading, as it suggests that you can hold n fixed when you let m go to infinity...which would imply that the sequence is constant, i.e. of the form v,v,v,v,v,..., and I'm pretty sure all of those are convergent.

A Hilbert Space is the sum of the multiplicative inverses of all known vectors that are real-valued scalar functions.
R[2]\prod[/1/x]

You're not making sense.

 

[zsk786]

Hey thanks guys this really help . I really appreciate it

 

[Cryxic]

I was wondering what is Hilbert Space exactly?
I read the Wikipedia page, but its one of those situations u understand what your reading but don't full grasp the concept.
I was just hoping someone could explain it to me.

Here goes nothing...

Hilbert space is a vector space.

A vector space is an algebraic structure that contains objects called vectors (in quantum mechanics, these become the wavefunctions), which display the properties of vector addition and scalar multiplication.

An algebraic structure is one or more sets that exhibit closure under one or more operations. An example would be the set of all real numbers under addition. If you add two real numbers, the third element will always be a real number. However, the set of all natural numbers (ie. positive integers) do not exhibit closure under, say, subtraction. 5 - 3 is 2 but 3 - 5 is -2, and -2 is not a natural number. So the set of all natural numbers cannot be an algebraic structure under subtraction (but can be algebraic structure under addition).

An example of a vector space could be the ordinary vectors that you learn about in elementary physics (these are rank 1 tensors), but it could also be the set of all 2x2 matrices. The vector space itself comes with a bunch of axioms for its vectors (rules for how vectors should behave), like invertibility, identity, and associativity. As a vector space, Hilbert space covers both real and complex numbers, and so the wavefunctions (vectors) that live in Hilbert space must be complex-valued.

It's all quite simple, really.