Special Properties of Hilbert Spaces?

[WWGD]

Hi All,
AFAIK, the key property that separates/differentiates a Hilbert Space H from your generic normed inner-product vector space is that , in H, the norm is generated by an inner-product, i.e., for every vector $v$, we have$||v||_H= <v,v>_H^{1/2}$, and a generalized version of the Pythagorean theorem is satisfied (Polarization Identity, Parallelogram Law). Why is this so special, and, what is it this property gives rise to that is not the case with more general inner-product normed vector spaces?

 

[mathman]

The most important distinction is that a Hilbert space is complete in the norm, i.e. every Cauchy sequence has a limit in the space. Note that the nom is defined by the square root of the inner product.

 

[WWGD]

But isn't any metric/normed space as good as a complete space anyway, since one can always do a metric completion of the space? And , yes, I did define the norm as the square root of the inner-product, I don't understand your comment in that respect.

 

[dextercioby]

With regards to the first post, you've got it a little wrong, or definately not clear, see the exact comment in the post below.

Mathematics tells us that a pre-Hilbert space is a pre-Banach space and this is just a consequence of algebra (nothing topological here), once you define the norm in the pre-Hilbert space in terms of its scalar product. But any pre-Banach is space is automatically a metric space, again this follows from algebra, once you define the metric with respect to the norm, thus with respect of the scalar product which originated the norm, hence this chain of spaces:

$$\{\mbox{pre-Hilbert spaces}\} \subset \{\mbox{pre-Banach spaces}\} \subset \{\mbox{metric spaces}\}$$

The completion theorem already brings you into the realms of topology. A theorem of completion is valid for all types of spaces and you can only issue one proof, not 3, the other follow automatically.

 

[dextercioby]

Hi All,
AFAIK, the key property that separates/differentiates a Hilbert Space H from your generic normed inner-product vector space is that , in H, the norm is generated by an inner-product, i.e., for every vector $v$, we have$||v||_H= <v,v>_H^{1/2}$, and a generalized version of the Pythagorean theorem is satisfied (Polarization Identity, Parallelogram Law). Why is this so special, and, what is it this property gives rise to that is not the case with more general inner-product normed vector spaces?

More specifically, it should have read: the key property that separates/differentiates a Hilbert Space H from your generic normed inner-product vector space is that H is complete with respect to the metric induced by the norm which in its turn was induced by the scalar product.

 

[WWGD]

With regards to the first post, you've got it a little wrong, or definately not clear. A pre-Hilbert space is a pre-Banach space and this is just a consequence of algebra (nothing topological here), once you define the norm in the pre-Hilbert space in terms of its scalar product. But any pre-Banach is space is automatically a metric space, again this follows from algebra, once you define the metric with respect to the norm, thus with respect of the scalar product which originates this chain of spaces.

$$\{\mbox{pre-Hilbert spaces}\} \subset \{\mbox{pre-Banach spaces}\} \subset \{\mbox{metric spaces}\}$$

The completion theorem already brings you into the realms of topology. A theorem of completion is valid for all types of spaces and you can only issue one proof, not 3, the other follow automatically.

I am not sure I get your point; as far as I know, a pre-Hilbert space is a normed inner-product vector space , whose norm originates from an inner-product. Once we have a norm, this determines the topology ; we have a metric topology originating from the open sets of the metric defined by d(x,y):=||x-y|| . So I don't see how to define a pre-Hilbert space without appeal to the underlying metric, and the metric implies a topology. How do we define a Hilbert space without referring to the metric/topological properties?

 

[WWGD]

More specifically, it should have read: the key property that separates/differentiates a Hilbert Space H from your generic normed inner-product vector space is that H is complete with respect to the metric induced by the norm which in its turn was induced by the scalar product.

But isn't any metric space as good as a complete space? Actually, in, say, L^p spaces for $p \neq 2$ , none
of the norms are induced by an inner-product , i.e., there are no inner-products < , >_p so that || f ||_p =
<f,f>_p for the standard norm ||f||_p := \Int (f^p)^{1/p}. This property is unique to Hilbert spaces. The completeness
of a metric is not a special property, I believe, because any metric can always be made complete (while, of
course, preserving the properties of the space).

 

[dextercioby]

I believe we can't see eye to eye here. The properties from the definition of a pre-Hilbert space don't include the topology. The metric topology comes as a consequence, once you define the metric directly from the scalar product or indirectly through the norm. Actually, you don't need to give a pre-Hilbert space the metric topology. You can give it the so-called weak topology which only includes scalar products without appealing to the induced norm/metric. A comment in post #7 is interesting. It addresses the existence of (pre-)Banach spaces which are not (pre-)Hilbert.

 

[WWGD]

We may be using different definitions; different authors have different layouts/assumptions. Still, re the original question: what do we get in a Hilbert space that we do not get in a regular inner-product vector space as a result of having a norm induced by an inner-product so that the norm is either complete or not ? i.e., in your layout, Dexter, what specially property do we get in Hilbert spaces that we do not get on other inner-product normed spaces?

 

[homeomorphic]

If you are going to say it's not important that it's complete because we can always make it complete, that seems to point to the conclusion that the important thing is that it is possible to complete it, thus reaping the benefits of completeness. However, you still have to keep in mind that all your theorems involving completeness are working in the bigger space, so that you are forced to refer to the bigger space to get things to work and not just work in the smaller space with the understanding that the completion takes care of everything. It's analogous to asking what's the big deal about the real numbers? We can just take the rationals and complete them. Well, that IS the real numbers.

So, what are the benefits of completeness? Taking projections is one of the big things, and the whole theory, really (the spectral theorem involves integrating projection operators, whose existence is provided for by completeness, for example). I haven't thought through exactly where it's needed at every point in the theory, though.

The key point, involving the construction of a minimizing vector going from a point to a subspace, involves the parallelogram identity to show that the sequence you construct is Cauchy and completeness to then get that it converges. This is how projection operators to a subspace are defined. Add the minimizing vector to the point and it takes it to its projection. Once you have your hands on that, a lot more things follow.

 

[dextercioby]

@homeomorphic, if I'm not mistaking the uniqueness of the projection is needed to prove the Riesz-Fischer representation theorem for (anti-)linear bounded functionals on Hilbert spaces. So if completeness fails, then R-F would fail, right?

 

[Stephen Tashi]

that we do not get in a regular inner-product vector space as a result of having a norm induced by an inner-product

Does that phrase refer to a space with a norm and an inner product such that the norm is not "induced" by the inner product? If so, we need a clear definition of the relation "not induced by".

 

[homeomorphic]

if I'm not mistaking the uniqueness of the projection is needed to prove the Riesz-Fischer representation theorem for (anti-)linear bounded functionals on Hilbert spaces. So if completeness fails, then R-F would fail, right?

I think so. The proof I have in mind definitely uses the existence of projections (onto the kernel and its orthogonal complement). There may be other proofs, but it should be possible to come up with a counter-example if you lack completeness.

 

[WWGD]

Does that phrase refer to a space with a norm and an inner product such that the norm is not "induced" by the inner product? If so, we need a clear definition of the relation "not induced by".

I guess we need to have a predefined norm ||.|| we will work with. Then:

This means that there is no inner-product < , > we can define ( meaning an operator defined on $V \times V \rightarrow \mathbb R$ satisfying the axioms of an inner-product so that <v,v>^{1/2} =||v|| )

For example, if in the L^p spaces we use the norm $||f||_p := (\int f^p)^{1/p}$ , we can show there is no inner-product <,> we can define so that $<f,f>^{1/2} = \int f^p)^{1/p}$ . One way of showing this is using the result that there is an inner-product that gives rise to the norm iff the norm satisfies the polarization identity.

 

[WWGD]

If you are going to say it's not important that it's complete because we can always make it complete, that seems to point to the conclusion that the important thing is that it is possible to complete it, thus reaping the benefits of completeness. However, you still have to keep in mind that all your theorems involving completeness are working in the bigger space, so that you are forced to refer to the bigger space to get things to work and not just work in the smaller space with the understanding that the completion takes care of everything. It's analogous to asking what's the big deal about the real numbers? We can just take the rationals and complete them. Well, that IS the real numbers.

.

But I think this ends up being a semantic issue; the Hilbert space is the completion ( if necessary) of the inner-
product space whose norm is induced by the inner-product. Yes, we would have to check that this property
of the norm is preserved under the completion, but I think that is not too hard. So the bigger space here is the
Hilbert space; the smaller one is the pre-Hilbert space. But then this depends on where/how you start defining terms.

And, yes, I had forgotten the special property of the existence of a unique shortest distance between a point p and a subspace S , this being reached between p and its orthogonal projection into S.

Dextercioby: Which of the versions of R-F are you referring-to ? Is it the completeness of L^p

EDIT: It would seem, from the context of the discussion that you are referring to the Riesz
representation theorem whereby every functional in H* can be represented/described by the inner-product against a fixed element x.

 

[mathman]

A simple example may help clarify the issue.
Let A=set of all finite series. An inner product and resultant norm can be defined on this space. A is not a Hilbert space and needs to include infinite series which have the property of finite sum of terms squared to be a Hilbert space.

 

[lavinia]

Hi All,
AFAIK, the key property that separates/differentiates a Hilbert Space H from your generic normed inner-product vector space is that , in H, the norm is generated by an inner-product, i.e., for every vector $v$, we have$||v||_H= <v,v>_H^{1/2}$, and a generalized version of the Pythagorean theorem is satisfied (Polarization Identity, Parallelogram Law). Why is this so special, and, what is it this property gives rise to that is not the case with more general inner-product normed vector spaces?

The inner product allows the idea of projecting one function onto another.
This can not be done for the other norms.

There are many applications of this.

- In probability, the inner product of two square summable (mean zero)random variables is their covariance. Orthogonal random variables are in most applications, independent.

- a complete set of orthogonal norm 1functions can be used as a"basis" for the Hilbert space and the cosine of the angles of a function with this basis gives you a series approximation to the function as a linear combination of the basis functions(possibility infinite combination). An example is the Fourier series.

 

[WWGD]

A simple example may help clarify the issue.
Let A=set of all finite series. An inner product and resultant norm can be defined on this space. A is not a Hilbert space and needs to include infinite series which have the property of finite sum of terms squared to be a Hilbert space.

So you are saying that the set of finite sequences, or , equivalently, the subset S of l^2 of sequences that are eventually zero are a Pre-Hilbert space, which becomes a Hilbert space after it is completed metrically into l^2?

I guess , e.g., we can have a sequence {a_n} with {a_n} zero after the n-th term, and this sequence would not
converge to a sequence with finitely-many terms.

Maybe I am missing something, but, what important property do we get in a Hilbert space that we do not get
in either a Pre-Hilbert space, or in a normed inner-product space that is not a Pre-Hilbert space? Is it the
projection operator that , given a subspace S and a point p, assigns to p a unique point p' so that ||p-p'||
is minimal. Is there anything else, any other major property?

 

[WWGD]

The inner product allows the idea of projecting one function onto another.
This can not be done for the other norms.

There are many applications of this.

- In probability, the inner product of two square summable (mean zero)random variables is their covariance. Orthogonal random variables are in most applications, independent.

- a complete set of orthogonal norm 1functions can be used as a"basis" for the Hilbert space and the cosine of the angles of a function with this basis gives you a series approximation to the function as a linear combination of the basis functions(possibility infinite combination). An example is the Fourier series.

But there are inner-product spaces that are not Hilbert spaces. Do you mean the fact that the inner-product comes from the norm?

And , is the existence of a maximal orthonormal set that becomes a Schauder basis unique to Hilbert spaces? Why is norm 1 special, can't we rescale any basis so that all basis vectors have norm 1? I think the Fourier series comes from the orthogonal projection of a function onto the span of the bases.

Sorry if I am being dense, but I just don't get the points here.

 

[lavinia]

But there are inner-product spaces that are not Hilbert spaces. Do you mean the fact that the inner-product comes from the norm?

And , is the existence of a maximal orthonormal set that becomes a Sc
hauder basis unique to Hilbert spaces? Why is norm 1 special, can't we rescale any basis so that all basis vectors have norm 1? I think the Fourier series comes from the orthogonal projection of a function onto the span of the bases.
Sorry if I am being dense, but I just don't get the points here.

- If the inner product space is complete then it is a Hilbert space by definition.

-Completeness allows Cauchy sequences to converge.

As has already been said, completeness together with the inner product leads to many results.

For a Hilbert space with a countable complete basis, completeness guarantees that the Fourier series converges. This allows one to express a function in terms of an orthonomal basis.( In general this may be an infinite linear combination hence the need for completeness.)

BTW: An orthonomal basis is used because convergence of the sum of the squares of Fourier coefficients easily implies that the Fourier series converges.

If a countable basis is complete, I would imagine that it can always be adjusted to be orthonormal using the Gramm-Schmidt process inductively.

 

[WWGD]

- If the inner product space is complete then it is a Hilbert space by definition.

-Completeness allows Cauchy sequences to converge.

As has already been said, completeness together with the inner product leads to many results.

For a Hilbert space with a countable complete basis, completeness guarantees that the Fourier series converges. This allows one to express a function in terms of an orthonomal basis.( In general this may be an infinite linear combination hence the need for completeness.)

BTW: An orthonomal basis is used because convergence of the sum of the squares of Fourier coefficients easily implies that the Fourier series converges.

If a countable basis is complete, I would imagine that it can always be adjusted to be orthonormal using the Gramm-Schmidt process inductively.

I think you need more than completeness of an inner-product space, otherwise , any Banach space with an inner-product would be a Hilbert space. But not all Banach spaces with inner-products are Hilbert paces; I think, e.g., C[0,1] with sup norm and $<f,g>:= \int_{[0,1]} fg$ is not a Hilbert space.

And you need more than completeness and inner-products, because this is just an inner-product Banach space. Again, C[0,1] is an example where you have completeness and an inner-product but not a Hilbert space. So you must be using the fact that the norm is derived from the inner-product, or else C[0,1] would have all those properties.

Besides, are you saying in every Hilbert space the Fourier series of every function converges ( and converges how; in norm, measure, etc. ). I think not all functions in , e.g., $\mathbb R$ have Fourier series that converge.

Ultimately, I think having a projection operator giving a unique shortest distance is what allows convergence of Fourier series; the Fourier expansion is the projection of the function into the space generated by the ortho basis.

 

[lavinia]

I think you need more than completeness of an inner-product space, otherwise , any Banach space with an inner-product would be a Hilbert space. But not all Banach spaces with inner-products are Hilbert paces; I think, e.g., C[0,1] with sup norm and $<f,g>:= \int_{[0,1]} fg$ is not a Hilbert space.

And you need more than completeness and inner-products, because this is just an inner-product Banach space. Again, C[0,1] is an example where you have completeness and an inner-product but not a Hilbert space. So you must be using the fact that the norm is derived from the inner-product, or else C[0,1] would have all those properties.

Besides, are you saying in every Hilbert space the Fourier series of every function converges ( and converges how; in norm, measure, etc. ). I think not all functions in , e.g., $\mathbb R$ have Fourier series that converge.

Ultimately, I think having a projection operator giving a unique shortest distance is what allows convergence of Fourier series; the Fourier expansion is the projection of the function into the space generated by the ortho basis.

A complete inner product space is a Hilbert space. The norm derives trivially from the inner product. Conversely a norm that satisfies the parallelogram law comes from an inner product.. The sup norm does not come from an inner product so the continuous functions on the unit interval with the sup norm are not a Hilbert space.

A Hilbert space is a metric space. Convergence means convergence in that metric in that space. The Fourier series of an element of a Hilbert space always converges to that element. The partial sums of the Fourier series form a Cauchy sequence and their distance to the function goes to zero. This may not be the same as point wise convergence. If a function's Fourier series does not converge then the function is not an element of the Hilbert space.

 

[homeomorphic]

Completeness is always with respect to a given norm. Evidently, you can be complete with respect to the sup norm, but not with respect to the inner product norm.

 

[WWGD]

But when you say that a complete inner-product space is a Hilbert space, you need to clarify what that means; you need to state that the norm originates from an inner-product. I don't think this is obvious. And having an ortho basis is also not unique to Hilbert spaces; any space with an inner-product can be given an ortho basis by, e.g., Gram-Schmidt.I think page 84 in here: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB4QFjAA&url=http://www.springer.com/cda/content/document/cda_downloaddocument/9780387952246-c4.pdf?SGWID=0-0-45-101626-p2187108&ei=TZ5iVL7OJuvCsAT-sIFQ&usg=AFQjCNEcavxT0K8w-gs6e2eLfDft29VJcA&bvm=bv.79189006,d.cWc

Here, Parseval's identity shows that for f in L^2, the partial sums of the Fourier series are bounded ( since the Fourier series for any function in an inner-product space are bounded by $||f||^2$. In an inn
er-product space (V, <,>) , with a countable ortho basis ${f_k}$
then the Fourier sum for any f in V is given by $\Sigma^{\infty}_{ i=1} <f,f_k>$ . But within L^2, this guarantees
that the Fourier series is bounded, i.e., the sum is finite.Given L^2 , there is an isometric isomorphism with l^2 , so that Parseval's inequality becomes an equality, and this
guarantees that the Fourier series of f in L^2 converge ( in norm) to f. And every Hilbert space is isomorphic
(maybe isometrically-isomorphic ) to l^2(K) for some set K. Ultimately, I think, another important property is that every closed subspace S has an ortho. complement, which allows the ortho projection to be the shortest distance between a point and S.

There is also the fact that every Hilbert space is isomorphic with its continuous dual ( of course, by cardinality reasons,
it cannot be isomorphic with its algebraic dual. ). Is this only true for Hilbert spaces?

I was just curious about what properties unique to Hilbert space accounted for all these properties.

Thanks to all for your comments, you helped me clarify my ideas/knowledge.

 

[WWGD]

Ah, I see, some people define an inner-product space as one where the norm derives from the inner-product.

 

[homeomorphic]

Ah, I see, some people define an inner-product space as one where the norm derives from the inner-product.

I think everyone just defines it as a space that has an inner product. If it has another norm, just replace it with the one from the inner product. So, I don't know that there's any difference in definitions here. Having an inner product gives you a norm. It might be a different norm than some other norm that you might put on the space. So, the space is going to be an inner product space with respect to one norm, and it might not be with respect to a different norm.

It's a strange statement to say that C[0,1] with the sup norm AND inner product given by the integral is not a Hilbert space. It IS a Hilbert space. It's just that it now has two different norms on it, instead of one. It's better to say that the sup norm doesn't come from an inner product. The space by itself without the inner product is something that gets turned into a Hilbert space or Banach space, maybe in more than one possible way. It's not Hilbert until AFTER you put the inner product on it. And it's not Banach until AFTER you put a norm on it. There could be more than one possible inner product or norm. Just as a set could become a group in many possible ways, unless it has a prime number of elements or something, so that we know that it's cyclic.

 

[WWGD]

Well, you're right, I meant to say and should have said that there is no inner-product that gives rise to the sup norm on C[0,1]. Then (C[0,1], Sup Norm) is not a Hilbert space. I stand corrected.
I meant to say that in some spaces, like in the case of C[0,1], there is a "standard/canonical" inner-product and a "standard/canonical" norm, and the two do not satisfy the norm being generated by an inner-product. If there is neither a canonical norm nor inner-product, then every space with an inner-product defined on it could trivially be made into a Hilbert space. So we ultimately start with a norm and then decide if the space with that norm is a Hilbert space.

 

[WWGD]

One more thing; I think the Cauchy-Schwarz inequality applies only to norms that come from inner-products.

 

[homeomorphic]

One more thing; I think the Cauchy-Schwarz inequality applies only to norms that come from inner-products.

Well, considering there is an inner product that explicitly appears in the inequality, I would guess you are probably right. :p

 

[WWGD]

Of course, what I meant is that you do not use the norm associated with the inner-product, the inequality may not hold.
Note of course that I did not state that the result may not hold in non-Hilbert cases, not that the result holds for Hilbert spaces. If you have an argument to this effect, I would gladly see it, if it as simple as you say.

 

[homeomorphic]

Oh, I see what you were getting at.

To get a counter-example, multiply the inner product norm by a small constant. It's still a norm. And note that if you inner product something with itself, you have equality in the Cauchy-Schwartz inequality. The shrunken square norm will be less than the regular square norm.

 

[homeomorphic]

And the counter-example is a norm that actually does come from some inner product. Just rescale the inner product.

 

[homeomorphic]

I should add that the same trick works for a norm that doesn't come from an inner product, if you want it to work for one of those. If it did obey Cauchy-Schwarz, you could re-scale it so that it didn't for any given inner product.

 

[WWGD]

O.K, good point.