User orbicular - MathOverflow

Answer by Orbicular for Differential of a Sobolev map between manifolds

2012-12-14T09:01:44Z

You have to extend J to be defined on your embedding space. Thus, your equation a priori depends on your chosen embedding. Of course the point here is to prove certain properties of your solutions that make it a posteriori independent of the embedding. For instance, Riviere often works with such embeddings and then shows regularity properties (such as harmonic maps from surfaces which are a priori only $W^{1,2},$ but then are shown to be Hölder).

In my opinion the main point here is the bigger bag principle. Classically for instance to solve the Dirichlet problem you use minimization of the Dirichlet functional. To capture the limit however requires you to use a complete space. Then to recover a usable solution you have to use regularity results. Notice however that you cannot use these regularity results if you do not have your bigger bag already.

It is similar in the holomorphic curve setting, where sometimes it is useful to consider suitable embeddings of the target manifold and work with Sobolev spaces with low regularity (such spaces are not intrinsically defined if the Sobolev space in question does not embed into the space of continuous functions). Usually in holomorphic curve theory one considers moduli spaces involving a fixed target space. Then it is feasible to work with target-depending moduli spaces as long as you can prove your wanted statement. Sometimes this is not possible since at some point you need more regularity (for instance in gluing).

Kuranishi structures vs polyfolds

2010-01-29T23:11:40Z

Moduli spaces of pseudoholomorphic curves do not carry the structure of a (compact) differentiable manifold in general (due to transversality issues). Nevertheless one would like to at least associate a fundamental class to the moduli space in question.
It looks like two approaches dominate: Kuranishi structures and polyfolds.
Both seem to be mammoth projects. Before diving seriously into one of them I may ask:
What are the advantages/drawbacks of these approaches? What is their motivation? Are they rigourosly proved? Are there reasonable alternatives?

Way to memorize relations between the Sobolev spaces?

2010-03-10T16:55:40Z

Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one can certainly feel lost. Are there certain tricks to memorize the (continuous and compact) embeddings between the different $W^{k,p}(\Omega)$ or into $C^{r,\alpha}(\bar{\Omega})$ ?

Is any Morse trajectory contained in a contractible open set?

2012-04-18T12:55:16Z

Suppose $f$ is a Morse function on a Riemannian Hilbert manifold $M$. Let $p_{\pm}\in \text{Crit}(f)$ be given and fix some $u:R\rightarrow M$ which is an integral curve of $-\nabla f$ connecting $p_-$ and $p_+.$ Is it true that there is an open contractible set $U\subseteq M$ containing the image of $u$?

Are there non-trivial locally trivial fiber bundles of Hilbert manifolds if the fiber is infinite-dimensional?

2012-05-02T11:54:12Z

It is well-known (see Kuiper's theorem) that every Hilbert space bundle over a manifold is trivial if the Hilbert space is truly infinite-dimensional.

Does the same hold true if one considers locally trivial fiber bundles instead and the fiber is assumed to be a truly infinite dimensional Hilbert manifold?

Is there a bound on the length of the longest Morse trajectory?

2010-10-04T10:08:08Z

Given a compact Riemannian manifold (with a fixed metric) and a Morse function on it (also fixed). Is there a bound (depending on the metric and the Morse function) on the length of the Morse trajectories? (You can assume the Morse-Smale condition if helpful.)

EDIT (In response to Dick's answer): The Morse function and the metric are fixed. I am just looking for something like $\int_{-\infty}^{\infty}\| \nabla f(\phi_t(p))\|dt\leq C$ and $C=C(f,g)$ where $f$ is the Morse function in question, $g$ stands for the metric and $\phi$ denotes the flow of $-\nabla f$. Note here that the constant is independent of the starting point $p$, as it is easy to see that such a constant additionally depending on $p$ exists (you use the hyperbolicity of $\nabla f$ to deduce exponential convergence towards a critical point). Furthermore it is also easy to see that the above integral is bounded if you include a $2$ in the exponent of the norm (a.k.a. $L^2$), as $\| \nabla f(\phi_t(p))\|^2=-\frac{d}{dt}f(\phi_t(p)).$

EDIT2 (In response to Bill's answer): I changed the statement to make it abundantly clear that the bound may depend on the metric and the Morse function. Maybe I was somewhat unclear in my formulation - sorry for that.

Gluing in Morse homology for Hilbert manifolds

2011-06-29T19:12:01Z

Is there any reference for gluing in the context of Morse homology on Hilbert manifolds?

Gluing is pretty standard in Morse homology for finite-dimensional manifolds. Unfortunately, in the infinite-dimensional case the sources I know avoid gluing. For proving that the Morse boundary operator squares to zero Abbondandolo-Majer use either an argument involving cellular filtrations or the graph transform method.

Unfortunately, in the context I want to consider, namely proving that some Morse-like theory is isomorphic to singular homology, both of these approaches do not work. (The latter approach works for showing that my Morse-like theory has a boundary operator that squares to zero, though.)

Reading through the finite-dimensional case ("Morse homology" of Schwarz) I realized that there are a lot of arguments which make it difficult to generalize the gluing procedure to cases where the target is not locally compact. For instance, in a lot of indirect arguments, the Rellich compact embedding theorem is used, which fails if the target is infinite-dimensional.

So, is there any instance where gluing in an infinite-dimensional context is proved? Are there certain obstacles not yet overcome?

cotangent bundle symplectic reduction and fibre bundles

2010-01-24T15:34:51Z

Suppose a compact Lie group $G$ acts on a manifold $M$ with only one orbit type $G/H$ ($H$ denotes the stabiliser group). Then the manifold $M$ becomes a fibre bundle over the quotient manifold $X:=M/G$ with typical fibre $G/H$ and structure group $G$.
On the one hand one could look at the cotangent bundle $T^* X$ of the quotient (which carries a natural symplectic structure).
On the other hand consider the lifted action of $G$ on the cotangent bundle $T^* M$ with moment map $\mu: T^* M\to \mathfrak{g}^* .$ The symplectic quotient $T^* M//G:=\mu^{-1}(0)/G$ inherits the structure of a symplectic manifold. Here comes the question: Are $T^* X$ and $T^* M//G$ (canonically) symplectomorphic?

Answer by Orbicular for Elliptic pseudodifferential operator estimate

2011-12-02T11:39:52Z

The answer to your question is no. Take any non-injective operator $P.$

the free loop space fibration is a locally trivial fiber bundle - reference?

2011-04-28T18:44:40Z

Let $Q$ be a compact Riemannian manifold. Then $\Lambda Q\rightarrow Q,$ $\gamma\mapsto \gamma(0)$ can be shown to be a locally trivial fiber bundle of Hilbert manifolds. Here, $\Lambda Q$ denotes the space of maps $S^1\rightarrow Q$ of Sobolev class $W^{1,2}.$

Question: Who proved it first? Is there an appropriate reference?

I once read it attributed to Klingenberg, but didn't find the proof (nor the statement) in the corresponding reference. I only know a proof due to Abbondandolo/Schwarz, but they claim no originality when asked.

Answer by Orbicular for Yang Mills gradient/heat flow on 4-torus

2011-11-18T17:07:13Z

I think you should just take a look at the following paper:

http://arxiv.org/PS_cache/arxiv/pdf/1103/1103.0845v1.pdf

(In particular it should exponential convergence.) Should there still be question, you might post them!

Answer by Orbicular for Eqivalency of two norms on the symmetric two tensor-fields on a compact Riemannian manifold.

2011-11-10T07:06:11Z

We have $\|D^2h-DD^\ast h\| \le \|D-D^*\|_{L^\infty(M,T^\ast M\otimes End(T^\ast M\otimes TM))} \|Dh\|,$ proving the uniform equivalence of the metrics. Notice that the difference of two connections is an ordinary tensor field, whose $L^\infty$-norm is bounded due to the compactness of $M$.

Another way to see this is that both metrics are locally equivalent to the metric $\xi\mapsto \|\nabla^2\xi\|+\|\xi\|,$ where now $\nabla$ refers to the "flat" connection and the $L^2$-norm refers to the local Lebesgue measure. Since $M$ is compact it can be covered by a finite number of such neighborhoods and in particular both norms are equivalent.

EDIT: I will keep the stuff I wrote at first (I thought of $D^*$ as a connection as well, which is not true, as Brian pointed out), but the correct answer involves elliptic regularity. I will consider the following simpler version (which also implies the general case, but once this simpler case is understood the more complicated case becomes easy.). Let $\nabla$ be a connection on $M.$ Then the following two norms are equivalent ($f\in C^\infty(M)$):
1. $\|f\|_{W^{2,2}}, $
2. $\|\Delta f\|_{L^2}+\|f\|_{L^2}.$
Clearly the second norm is dominated by the first. The other inequality follows from the elliptic estimate
$\|D^2 f\|_{L^2}\leq \|\Delta f\|_{L^2},$ valid for $f\in C^\infty_0(R^n),$ which is actually trivial to prove.

Answer by Orbicular for norm of n-th covariant derivative of smooth function

2011-11-10T18:04:38Z

From the perspective of analysis it is usually more convenient to define your favorite class of function spaces via local coordinates and a partition of unity. More precisely consider your favorite (compact manifold) and choose a covering $U_i$ together with charts $\phi_i:B_1\rightarrow U_i,$ where $B_1$ denotes the unit ball in your model space and such that $\phi$ extends to the closures, and a partition of unity $(\chi_i)$ subordinate to your covering. The you define

$\|f\|_{W^{k,p}(M)}:=\sum_i \|\phi_i^\ast (\chi_if)\|_{W^{k,p}(B_1)}$ for $f\in C^\infty(M).$

This norm is independent of the choices as long as $M$ is compact and its closure is a Banach space. Vector bundles can be treated similarly. Of course this defines a norm which equivalent to choosing a metric and a connection $\nabla$ and to consider

$\|f\|:=\|\nabla^k f\|{L^p}+\|f\|{L^p},$

as in Spiro's answer.

Elliptic regularity in Sobolev spaces of negative order

2011-10-25T19:11:09Z

Consider 1 < $p<\infty$ and an integer $k$. Does interior elliptic regularity for the Laplacian also hold in the Sobolev space $W^{k,p}$ of negative order?

More precisely I am interested in the following question: Let $u\in W^{-1,p}(R^n)$ be a distributional solution of $\Delta u=Su,$ where $S$ is smooth. Is it then true that $u$ is smooth?

Answer by Orbicular for Dimension of moduli space in Lagrangian Floer homology

2011-10-21T19:32:26Z

For a general submanifold P the problem you pose is not a Fredholm problem. In particular, you cannot expect your moduli space to be finite-dimensional. This has to do with admissible boundary conditions for the Cauchy-Riemann equation (buzzword: totally real boundary condition).

I am not so sure about the dimension of the moduli space. However I do know that the dimension can be read off from the Fredholm index of the linearized problem. I would expect some Maslov index to appear.

Elliptic estimates and regularity of the $\overline{\partial}$-operator with totally real boundary conditions in $W^{1,p},$ $1
2011-10-21T09:40:02Z

Let $\Omega$ be an open subset of the upper half-plane in the complex plane. I am considering the following problem:

(1) $\overline{\partial}u=f,$ $\textrm{Im} f=0$ on the real line for maps complex-valued maps on $\Omega.$

Here, $\overline{\partial}$ denotes the classical Cauchy-Riemann operator. Usually one considers (1) as posed in the space (of complex-valued maps lying in) $W^{1,p}$ for $p>2$ because by Sobolev embedding elements of $W^{1,p}$ are actually continuous and the boundary condition is then well-defined. One then has elliptic regularity in the sense that if $u\in W^{1,p}(\Omega)$ solves (1) for $f\in W^{k,p}(\Omega),$ then $u\in W^{k+1,p}_{loc}(\Omega)$ with corresponding estimates. (By usually I mean the references on J-holomorphic cures like Salamon-McDuff's books or the book by Abbas-Hofer.)

I am actually looking for the corresponding statement for $p\in(1,2].$ Namely consider the space $W^{1,p}(\Omega)\times W^{1,p}_0(\Omega)$ (now both spaces consist of real-valued maps and correspond to real and imaginary part), where $W^{1,p}_0(\Omega)$ is the closure of compactly-supported smooth $u\in C^{\infty}(\Omega),$ s.t. $\textrm{supp } u$ is disjoint from the real axis. Is it then true that a weak solution $u\in W^{1,p}(\Omega)\times W^{1,p}_0(\Omega)$ of $\overline{\partial}u=f$ with $f\in W^{k,p}$ is in fact in W^{k+1,p}?

As I already mentioned the literature I came across only treats the case $p>2.$ Notice also that it does not suffice to use regularity theory for the Laplace operator, since one only gets local regularity for the real part (i.e. no regularity up to the boundary).

cokernels of semi-Fredholm operators

2011-10-18T19:54:30Z

I did not find a reference for the following question, so I will pose it here. I think the answer should be elementary.

Let $F:X\rightarrow Y$ be a semi-Fredholm operator between Banach spaces, i.e. $\ker F$ is finite-dimensional and the image of $F$ is closed. Usually one wants to argue that $F$ is even Fredholm, thus exhibiting $\mathrm{coker} F:=Y/\mathrm{im} F$ as finite-dimensional. The usual argument involves the annihilator of the image of $F,$ which is easily seen to be isomorphic to $(\mathrm{coker}F)'$. More precisely, the annihilator of the image is connected to weak solutions of the (formally) adjoint operator. Once one has shown that the annihilator of the image is finite-dimensional (usually via ellipticity), it follows that the annihilator of the image is isomorphic to the cokernel of $F$.

But I have also read sources where the cokernel is directly related to the annihilator of the image. Is it true that $(\mathrm{im} F)°\cong \mathrm{coker}F$ for the situation described above? It is certainly true if $X$ and $Y$ are Hilbert spaces.

reference for the slice theorem for Banach Lie group actions on Banach manifolds

2011-09-26T12:27:42Z

I am looking for a reference treating the slice theorem for Banach Lie group actions on Banach manifolds, i.e. proving that a smooth, free and proper action of a Banach Lie group $G$ on a Banach manifold $M$ with embedded orbits ensures that the quotient $M/G$ inherits the structure of a Banach manifold, s.t. $M\rightarrow M/G$ becomes a $G$-principal bundle.

I know this is briefly treated in Bourbaki's "Lie groups and Lie algebras". Unfortunately, in the proof they refer to the Bourbaki book "differentiable and analytic manifolds" which is not available to me.

Could anyone please provide me with a reference?

Is it possible to make the principal bundle projection map totally geodesic?

2011-09-23T09:44:06Z

Let $G$ be a compact (connected) Lie group. Suppose that a $G$-principal bundle $\pi:P\rightarrow Q$ is given.

Is it always possible to equip $P$ and $Q$ with Riemannian metrics, s.t. $\pi$ is totally geodesic? Notice that $g_P,$ the metric on $P,$ does not have to be $G$-invariant and $\pi$ does not have to be a Riemannian submersion.

on a property of functions of quadratic growth

2011-06-07T14:32:55Z

I want to verify the following claim that I found in some paper. Suppose f is a smooth real-valued function on the real line satisfying $f'(x)x-f(x)\ge x^2$ for all x. Then there is a constant C, s.t. $f(x)\ge \frac{x^2}{2}-C$ for all $x.$

The connection with the title is that the authors claim that any function satisfying the first inequality is of quadratic growth, i.e. satisfies the second inequality.

continuity of extension of maps along curves

2011-02-13T12:06:31Z

Let $a\le b$ and $k\ge 0$ be given and fixed. Let furthermore $x$ and $y$ denote two different elements of a Hilbert space $H$. Suppose $u:\mathbb{R}\rightarrow H$ is a $C^k$-embedding connecting $x$ and $y$, s.t. the derivatives up to order $k$ vanish at infinity and $f:\mathbb{R} \rightarrow \mathbb{R}$ a given $C^k$-map with support in $[a,b].$ Then it it possible to construct a $C^k$map $\tilde{f}:H\rightarrow \mathbb{R}$ satisfying $\tilde{f}(u(s))=f(s)$ for any $s\in \mathbb{R}.$ (This can be done using a tubular neighborhood and a suitable cutoff function.)

Is it possible to make the map $f\mapsto\tilde{f}$ continuous? The domain of this map should be the space of $C^k$-embeddings (as above) times $\mathbb{R}$-valued $C^k$- functions supported in $[a,b]$. And the codomain should be the space of $C^k$-maps on $H$.

Do extracted weak $H^{1,2}$-limits and $C^0$-limits coincide?

2011-05-18T14:51:03Z

Let $I$ be a bounded interval and consider a sequence $(u_k)$ in $H^{1,2}(I)$ (usual Sobolev space). Suppose furthermore, that the sequence $(u_k)$ is bounded in $H^{1,2}(I)$. Then, by Rellich, we can extract a subsequence, still denoted by $u_k,$ s.t. $u_k$ converges to some $\bar{u}$ in $C^0(I)$. Furthermore, by weak compactness of bounded sets in $H^{1,2}(I)$ we can select a subsequence, s.t. $u_k$ converges to some $u$ weakly in $H^{1,2}(I)$. Thus, $u_k$ converges to $\bar{u}$ in $C^0$ and weakly to $u$ in $H^{1,2}$.

Do these limits coincide, i.e. is it true that $u=\bar{u}$?

cutting off $H^{1,2}$-functions in the image

2011-04-02T19:45:43Z

Suppose $\Omega$ is a bounded domain in $\mathbb{R}^n.$ Let $w\in H^{1,2}$ (standard Sobolev space, order 1, integrability 2) and $L>0$ be given. Is it then true that the function $w_L:=\min (L,w)$ is also in $H^{1,2}?$

I found this assertion in the book "Riemannian geometry and geometric analysis" of Jost, in the section concerning higher regularity of harmonic maps. There, one already knows that $f$ (the continuous weakly harmonic map) is in $H_{loc}^{1,4} \cap H^{2,2}_{loc}$ and considers $w:=|Df|^2.$

Now, the function $w_L$ can also be written as $x\mapsto w(x)\chi_L(w(x)),$ where $\chi_L$ denotes the characteristic function of the set consisting of all x s.t. $x\le L$. Considering the distributional derivative of $w_L$ gives you something involving a delta function(al), which is not in $L^2.$

Any suggestions? What am I doing wrong?

Transversality in Morse theory for the (perturbed) geodesic action functional

2011-02-11T14:43:47Z

I am interested in Morse homology on the loop space of a given compact (Riemannian) manifold. A small perturbation renders the geodesic action ("energy") functional Morse. Now I am interested in the Morse-Smale property, i.e. for any critical points x and y the unstable manifold of x intersects the stable manifold of y transversally.

Could anyone please provide a reference that a generic choice of metric on the loop space yields the Morse-Smale property? (Notice that the correct choice of perturbations of the metric is part of the problem.) I have difficulties finding an appropriate reference for this.

There seem to be two obvious ways to realize Morse-Smale transversality in this setting:
1. The abstract way: Here one considers a given Hilbert manifold with a metric. The space of perturbations consist of (some class of) metrics which are uniformly equivalent to the given one. This is for instance the approach followed by Abbondandolo/Majer: "Lectures on Morse homology for infinite-dimensional manifolds". The problem with this reference is that their space of perturbations is too big - the space in question is not separable. In particular the Sard-Smale theorem, which is crucial in this setting, cannot be applied. I have difficulties in writing down a separable Banach space of perturbations which is still enough to provide surjectivity of the linearized "master section".
2. The concrete setting: Obviously, it is not enough to consider metrics on the loop space which come from metrics on the base manifold. I do not know whether it suffices to consider metrics on the loop space which come from metrics on the base times $S^1.$ My problem is that the "master section" involves the gradient (w.r.t. the induced metric on the loop space) of the perturbed energy functional in question. I have no clue how to obtain a useful formula for its linearization.

So, could anyone please give me a hint about solving 1. or 2.? It is also possible that pursuing the paths 1. or 2. might not be a clever idea, in which case I would appreciate any advice.

Answer by Orbicular for Are infinite dimensional constructions needed to prove finite dimensional results?

2011-02-17T20:54:19Z

This one is analogous to an Ramras' answer: Donaldson's theorem from the 80s and the corresponding Seiberg-Witten stuff.

Answer by Orbicular for Are infinite dimensional constructions needed to prove finite dimensional results?

2011-02-17T20:27:14Z

Is it possible to prove the (un)stable manifold theorem for rest points of hyperbolic vector fields without using suitable (infinite dimensional) Banach spaces?
I am only aware of proofs that use certain spaces of curves or sequences which are infinite-dimensional.

Can Morse trajectories break if their first derivative is uniformly bounded?

2011-02-17T09:23:12Z

Consider a compact Riemannian manifold endowed with a Morse function f. Fix two critical points x and y of f. Whenever you have a sequence $u_k$ of Morse trajectories connecting x and y it might happen that the $u_k$ "converge" to a broken trajectory. Suppose there is a bound $\|u_k'(s)\|\le C$ independently of k and s. Can the sequence still break?

separability of a certain space of continuous functions, II

2011-02-02T17:22:39Z

This is a follow-up of a question of mine with a similar title. I am interested in Morse homology (on Hilbert manifolds), more specifically with "generic" perturbations of the metric tensor (under the heading of "transversality"). The space of perturbations to use should have the property of separability in order to apply the Sard-Smale theorem.

Now comes the question, which is supposed to help me in these matters: Let H be a separable Hilbert space and let B be the closed unit ball in H. Is the space C^b(B), the space of continuous bounded functions on the closed unit ball endowed with the sup-norm, a separable space?

The previous question of mine replaced B above with an open subset of the Hilbert space. Then the answer turns out to be NO. Notice that if H is finite dimensional, the answer is YES.

separability of a certain space of continuous functions

2011-02-01T21:21:34Z

Let $O$ be an open subset of the separable Hilbert space $H.$ Let $E$ be a separable Banach space. Is it true that $C^0_b(O;E),$ the space of bounded continuous maps $O\rightarrow E$, endowed with the $C^0$-norm, is separable? If YES, where can I find I proof of this fact?

Separability of the space of bounded continuous maps

2011-01-25T14:13:53Z

Let $O$ be an open subset of the separable Hilbert space H and $k\geq0$ . Consider $C_b^k(O, Sym(H))$, the space of k-times continuously differentiable maps with values in the bounded symmetric endomorphisms of $H$, bounded up to their k-th derivative. Equipped with the usual norm this space becomes a Banach space. Is this space separable, i.e. has a dense sequence?

I need this result for transversality theory in Morse theory, where the space above serves as a space of suitable perturbations. The separability is needed in order to aplly the Sard-Smale theorem.

Comment by Orbicular

2012-05-03T07:34:04Z

Dear Alberto, your comments answers my question. Thanks!

Comment by Orbicular

2011-12-13T16:01:36Z

Actually this was well-known a long time before the advent of deRham cohomology...

Comment by Orbicular

2011-12-07T21:57:43Z

google flow box theorem

Comment by Orbicular

2011-12-02T12:50:45Z

If your operator is injective then you can even estimate the H^1 -norm of u in terms of the L^2-norm of u. This is true because any injective operator with closed image satisfies an injectivity estimate.

Comment by Orbicular

2011-11-30T00:02:37Z

I do not understand why you should need Banach-space-valued functions. The pointwise norm of a section defines an ordinary function on M, in particular it is clear what to integrate.

Comment by Orbicular

2011-11-17T07:59:14Z

To elaborate on Matthew's answer: If the kernel of the adjoint is finite-dimensional, then it is isomorphic to the cokernel of the operator you started out with. You do not need Fredholmness of $L$. You need the $L$ has a closed image and that the kernel of $L^*$ is finite-dimensional.

Comment by Orbicular

2011-11-12T13:23:04Z

Thank you, Brian!

Comment by Orbicular

2011-11-11T11:34:42Z

The math does not display properly. I don't know why and would appreciate help.

Comment by Orbicular

2011-11-10T22:44:06Z

@Brian: The latter expression is a (0,4)-tensor. h is a (0,2) tensor and any covariant derivative adds a (0,1).

Comment by Orbicular

2011-10-25T20:33:52Z

@Spencer: S is a smooth function.

Comment by Orbicular

2011-10-25T20:28:18Z

I posted it on math.stackexchange.com, but did not get a useful answer. So I decided to try it here.

Comment by Orbicular

2011-10-21T17:50:39Z

@Sam: Thanks for the answer! In the version of Abbas/Hofer that I have (possibly not the latest one) they prove interior $L^p$-estimates ($p>1$) for the CR-operator. They only discuss regularity in $W^{1,p}$ for $p>2.$ Nevertheless I will accept your answer since I realized that I have stupidly missed that the corresponding statement is proved in Appendix B of the newer McDuff-Salamon treatise. They prove interior $L^p$-regularity ($p>1$) and reduce the boundary regularity (with vanishing imaginary part a.e.) to the interior regularity via Schwarz reflection.

Comment by Orbicular

2011-10-18T20:42:46Z

@Bill: I know, but that was not the question. The question was whether one can always identify the annihilator of $Y$ with $X/Y$. I assume this is false, since one can simply take some Banach space $Z$ which is not isomorphic to its dual and consider $Y=0$ and $X=Z.$ Am I correct?

Comment by Orbicular

2011-07-15T12:31:44Z

I think Lang treats this stuff in his "real and functional analysis" book

Comment by Orbicular

2011-07-04T08:58:54Z

Tim, you are correct. I deleted the comment.