Exploring Hilbert Space: What Makes It Confusing?

[ognik]

I see I am not the only one finds Hilbert confusing - because all it's properties seem so familiar. I have gathered together what I could find, please comment?

A Hilbert space is a vector space that:
Has an inner product:
• Inner product of a pair of elements in the space must be equal to it's complex conjugate
• Inner product of an element in the space with itself is greater than or equal to 0
• Inner product is linear in 1st element
Is linearly complete
Is usually infinite dimensional
Has to satisfy the triangle inequality

I also found this - The inner product has to be 0 only if one of the vectors I give it is 0. But what if the vectors are orthogonal?

The differences from a vector space, including the infinite dimension point, seem too subtle for me to grasp; what are they essentially?. Why is Hilbert space so important?

 

[Ackbach]

I see I am not the only one finds Hilbert confusing - because all it's properties seem so familiar. I have gathered together what I could find, please comment?

A Hilbert space is a vector space that:
Has an inner product:
• Inner product of a pair of elements in the space must be equal to it's complex conjugate

Not true as stated. You have to swap the order of the vectors in the inner product: $\langle x|y\rangle = \langle y|x\rangle ^*$.

• Inner product of an element in the space with itself is greater than or equal to 0
• Inner product is linear in 1st element

This varies by author. Physicists are universal (because of Dirac notation, which I just used above) in having the inner product linear in the second term, and only conjugate-linear in the first term. Mathematicians are wishy-washy about this, and vary from author to author. Some have it the same way as the physicists, and some the opposite. Know your author!

Is linearly complete

This is precisely what distinguishes an inner product space from a Hilbert space. A Hilbert space is a complete inner product space.

Is usually infinite dimensional
Has to satisfy the triangle inequality

I also found this - The inner product has to be 0 only if one of the vectors I give it is 0. But what if the vectors are orthogonal?

Orthogonality is defined by the inner product. I think this statement you found is talking about when, a priori, you know that the inner product of two vectors is zero. If neither of the two vectors is zero, then you inner product doesn't have to be zero - unless the two vectors are orthogonal. But you likely wouldn't know that in advance, unless you compute the inner product.

The differences from a vector space, including the infinite dimension point, seem too subtle for me to grasp; what are they essentially?. Why is Hilbert space so important?

So, from a vector space, you first introduce the inner product that has to satisfy certain properties. A vector space isn't necessarily an inner product space. Then, to achieve a Hilbert space, you complete the inner product space (add in all limit vectors).

Hilbert space is of immense importance. Quantum Mechanics (in the matrix Heisenberg/Schrödinger equation version) builds its entire theory on the structure of Hilbert space. (The Lagrangian formulation is a bit different.) In addition, I have heard one individual say that all of applied mathematics is just operator theory on Hilbert space. Perhaps a tad overstated, but not too far off.

 

[Deveno]

I have also heard it stated that the only Hilbert space really worth studying is $\ell^2$, up to isomorphism.

The goal is to be able to extend our geometric intuition of $\Bbb C^n$ to the vector space of all functions $\Bbb C \to \Bbb C$. Well, for starters, this space is "too big" and not-well-behaved enough to work in satisfactorily.

However, many of the functions we are actually interested in can be represented by power series. "Truncated" power series are just polynomials, and the space of all (complex) polynomials of degree $n$ or less is, for all intents and purposes, $\Bbb C^n$.

So, basically, we'd like to extend our intuition of "finite" linear combinations to "infinite" linear combinations. The problem is, an "infinite sum" doesn't make much sense...unless the sum "tends to something".

So, our first idea might be to limit ourselves to "convergent" infinite sums, or even better "absolutely convergent" infinite sums (the latter space is $\ell^1$).

However, for our "usual" dot product to have meaning in this context, we will need our "infinite sums" to be "square-summable". This is the "motivation" behind $\ell^2$.

On another tack, the expression:

$\int\limits_a^b f(x)\overline{g(x)} dx$

has many of the properties of an inner product. For us to use this as an inner product, a sufficient condition is that $f$ be "square-integrable" (for reasons having to do with the behavior of the integral with respect to limits, the Lebesgue integral is used-not "enough" functions are Riemann-integrable, and the resulting space isn't *complete*). This gives rise to the Hilbert space $L^2$, which is the "natural" domain to study Fourier series, and the related Fourier transform.

These are "the big guys" that dominate the theory. Because the existence of an inner product is sometimes "more than we need", sometimes we relax our condition to the existence of just a norm (every inner product defines a norm, but the converse is not necessarily so). This is still strong enough to induce a metric topology, so even though we may no longer be able to quantify "angle", we can still measure "how far apart" two vectors are.

In particular we have the intersection of 3 main areas of mathematics:

1. Algebra-from linear algebra and ring theory.
2. Real analysis (most forms of multi-dimensional analysis, even complex-valued, ultimately derive their truth from the construction of the real numbers).
3. Topology - usually in a highly "special" form, where we have many "separation" axioms to distinguish sets.

 

[ognik]

Thanks Ackbach, nice.

Checking, my understanding of completeness is:

"Completeness means that if a particle moves along a broken path (set of vectors with different directions, arranged head-to-tail) traveling a finite total distance, then the particle has a well-defined net displacement"

I think your words imply more than this?

Also I haven't encountered 'limit vectors' and couldn't find anything online?

Finally, while I am clearer on what it is and what is used for, I still can't grasp what specific aspect(s) of Hilbert space make it more useful (for say QM) - it's properties still seem familiar to me, for example compared with (my understanding of) vector space? Also for example, is vector space not complete?

 

[ognik]

I have also heard it stated that the only Hilbert space really worth studying is $\ell^2$, up to isomorphism.

Thanks also Deveno, also useful. I haven't encountered $\ell$ space before - it sounds to good to be true, to be able to assume that if we are in $\ell$ space, all series are absolutely convergent? Or do we still have to check that?

I read for $\ell^2$:
"A good way to think of an $\ell^2$-function is as a density function" and
"the $\ell^p$ spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces"

Also it seems that functions in $\ell^2$ are continuous? (Wouldn't that conflict with quantisation?)
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Guys, your knowledge is beyond impressive, but I think, for me at present, a deeper understanding of this will have to wait until a bit later, are there perhaps a few important practical properties I should be aware of? Something like the inner product is always normalised ...this year has become more of a preparation for the physics to follow next year...