User andr&#225;s b&#225;tkai - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:30:57Z http://mathoverflow.net/feeds/user/12898 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/131070/algorithm-to-find-exponential-map-of-differential-operators-acting-on-function/131075#131075 Answer by András Bátkai for Algorithm to find exponential map of differential operators acting on function András Bátkai 2013-05-18T18:19:57Z 2013-05-18T18:19:57Z <p>A partial answer: What you call the "exponential function" is the so-called flow semigroup, see <a href="http://www.fa.uni-tuebingen.de/research/publications/1999/one-parameter-semigroups-for-linear-evolution-equations/" rel="nofollow">Engel-Nagel</a>, Section II.3.28.</p> <p>Another reference on the <a href="http://en.wikipedia.org/wiki/Lie_derivative" rel="nofollow">Lie derivative</a> is the monograph by <a href="http://ejde.math.unt.edu/Monographs/02/abstr.html" rel="nofollow">Chicone and Swanson</a>.</p> http://mathoverflow.net/questions/130857/proof-that-l20-tx-l20-tx/130865#130865 Answer by András Bátkai for Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$ András Bátkai 2013-05-16T19:25:44Z 2013-05-17T09:04:28Z <p>To give you a reference: <a href="http://books.google.hu/books?id=NCm4E2By8DQC&amp;printsec=frontcover&amp;hl=de#v=onepage&amp;q&amp;f=false" rel="nofollow">Diestel-Uhl, Vector measures</a>, page 98, Chapter 4, Theorem 1:</p> <p>$$L^p(\mu,X)^\ast = L^q(\mu,X^\ast)$$</p> <p>if and only if $X^\ast$ has the Radon-Nikodym property with respect to $\mu$. </p> <p>Here $\mu$ is a finite measure and $p$ and $q$ as usual. Proof can be read there.</p> <p>ADDED: It should be noted that $L^q(\mu,X^\ast) \subset L^p(\mu,X)^\ast$ always holds, without any condition on the Banach space $X$. The proof of this is quite the same as in the scalar case.</p> http://mathoverflow.net/questions/129992/radon-nikodym-property-of-ell-infty/130000#130000 Answer by András Bátkai for radon-nikodým property of $\ell^\infty$ András Bátkai 2013-05-07T18:07:16Z 2013-05-07T20:11:24Z <p>See Proposition 1.2.9 in <a href="http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0086-0" rel="nofollow">Arendt-Batty-Hieber-Neubrander</a>. (I look at the first edition now) </p> http://mathoverflow.net/questions/125302/the-periodic-schrodinger-group/125313#125313 Answer by András Bátkai for The Periodic Schrödinger Group András Bátkai 2013-03-22T19:15:25Z 2013-03-23T08:42:21Z <p>Heuristically it is what Ryan Budney suggests, but of course it does not make sense as a power series, only for so called analytic vectors (which are dense though in this case).</p> <p>Usually you define this via the spectral theory of selfadjoint operators, see for example </p> <p>Weidmann: <a href="http://books.google.de/books/about/Linear_operators_in_Hilbert_spaces.html?hl=de&amp;id=_SDvAAAAMAAJ" rel="nofollow">Linear operators in Hilbert space</a>, Section 7.6. </p> <p>The construction to show that the spectral theorem in this case reduces to the operator being a Fourier multiplier is described in Chapter 10.</p> http://mathoverflow.net/questions/121563/on-exponential-formula/121576#121576 Answer by András Bátkai for On exponential formula András Bátkai 2013-02-12T10:14:22Z 2013-02-12T17:55:40Z <p>If you have an analytic semigroup (generated by a so-called <a href="http://en.wikipedia.org/wiki/Analytic_semigroup" rel="nofollow">sectorial operator</a>), then the answer is well-known and yes, even convergence rates are possible. This and generalizations were proved by many, see for example the summary and references in</p> <p><a href="http://www.springerlink.com/content/l4r551nj87143385/" rel="nofollow">M. Crouzeix, S. Larsson, S. Piskarev, V. Thomée: The stability of rational approximations of analytic semigroups, BIT 33 (1993), 74-84</a>, Theorem 3. </p> <p>In general it is possible to obtain convergence rates for "nice" initial values. A nice summary of the existing results and vast generalizations are available in this paper by <a href="http://arxiv.org/abs/1301.4406" rel="nofollow">Alexander Gomilko and Yuri Tomilov</a>.</p> <p>ADDED: After discussion with Delio, let me mention that the result (in Hilbert spaces) is implicitely contained in</p> <p><a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.jmsj/1240433924" rel="nofollow">H. FUJITA &amp; A. MIZUTANI,</a> "On the finite element method for parabolic equations. I; Approximation of holomorphic semi-groups," J. Math. Soc. Japan, v. 28, 1976, pp. 749-771.</p> http://mathoverflow.net/questions/111290/commutator-formula-in-infinite-dimensions/121349#121349 Answer by András Bátkai for Commutator formula in infinite dimensions András Bátkai 2013-02-09T19:33:31Z 2013-02-09T19:33:31Z <p>For the unbounded case, see</p> <p>Kato, Tosio <a href="http://www.ams.org/mathscinet-getitem?mr=538020" rel="nofollow">Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups.</a> Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), pp. 185–195, Adv. in Math. Suppl. Stud., 3, Academic Press, New York-London, 1978.</p> <p>The domain condition is that the square roots of $A$ and $B$ should have a dense intersection, then the convergence holds, to the exponential of the form sum of these operators.</p> http://mathoverflow.net/questions/109279/when-should-i-publish-my-results/109282#109282 Answer by András Bátkai for When should I publish my results? András Bátkai 2012-10-10T09:19:00Z 2013-02-05T08:18:25Z <p>Understanding old results and putting them into new perspective has its own very important value. You should not underestimate it.</p> <p>However, especially for a novice, it is difficult to judge this. Consult your reviewer and ask for advice. If an experienced and well-established mathematician sees value in your work, then you should try to publish it.</p> <p>Of course it is an important and difficult question where. Not all journals approve of this type of synthetizing papers. Here again, the help of an experienced established colleague is inevitable.</p> http://mathoverflow.net/questions/120198/generator-of-a-generated-c-0-semigroup/120231#120231 Answer by András Bátkai for Generator of a generated $C_0$ semigroup. András Bátkai 2013-01-29T16:49:36Z 2013-01-29T16:49:36Z <p>The semigroups you construct is in general only weak-* continuous. Are you looking for so-called implemented semigroups?</p> <p>See for example <a href="http://www.springerlink.com/content/hgx6qemft9wpu76h/" rel="nofollow">this</a> paper.</p> http://mathoverflow.net/questions/118619/robin-laplacian-in-unbounded-domains/118620#118620 Answer by András Bátkai for Robin-Laplacian in unbounded domains András Bátkai 2013-01-11T12:31:01Z 2013-01-11T12:31:01Z <p>For the case $p=2$, I would have a look at <a href="http://cantor.mathematik.uni-ulm.de/m5/arendt/publications/arendt-pub/short/2003-AreWar-LplRbnBndCndArbDmn.pdf" rel="nofollow">this</a> paper by Wolfgang Arend and Mahamadi Warma, and its follow-up papers: Potential Analysis 19: 341–363, 2003.</p> http://mathoverflow.net/questions/118284/schauder-estimates-for-higher-order-linear-elliptic-operator-on-manifold/118292#118292 Answer by András Bátkai for Schauder estimates for higher order linear elliptic operator on manifold András Bátkai 2013-01-07T16:09:00Z 2013-01-07T16:09:00Z <p>This is probably more of a comment: Section 3.2 in <a href="http://books.google.hu/books?id=mWojiHzg9bEC&amp;printsec=frontcover" rel="nofollow">Lunardi's book</a> contains a broad overview on higher order parabolic problems with lots of references, including the most important Hölder estimates. </p> <p>On manifolds, you should be able to extend these results using a finite number of coordinate charts (by compactness) as in <a href="http://www.math.utsc.utoronto.ca/gpde/notes/schmfld.pdf" rel="nofollow">these notes</a>. But, this is not a reference...</p> http://mathoverflow.net/questions/117668/new-grand-projects-in-contemporary-math/118265#118265 Answer by András Bátkai for New grand projects in contemporary math András Bátkai 2013-01-07T11:06:56Z 2013-01-07T11:06:56Z <p>The theory of nonlinear dispersive equations, hyperbolic conservation laws, etc., see</p> <p>Terece Tao's <a href="http://terrytao.wordpress.com/books/nonlinear-dispersive-equations-local-and-global-analysis/" rel="nofollow">book</a> on the subject, Jean Bourgains <a href="http://books.google.hu/books?id=FZ7-TFGS9LAC" rel="nofollow">book</a> or Helge Holdens co-authored <a href="http://www.math.ntnu.no/~holden/books.html" rel="nofollow">monographs</a>. </p> http://mathoverflow.net/questions/117767/generator-of-a-c-0-semigroup-restricted-to-a-subspace/117775#117775 Answer by András Bátkai for Generator of a $C_0$-semigroup restricted to a subspace András Bátkai 2013-01-01T10:32:09Z 2013-01-01T11:00:16Z <p>I do not think such semigroups have been <em>extensively</em> studied.</p> <p>I have seen such semigroups (and, more generally, evolution families corresponding to the non-autonomous problem) in</p> <p>A. Lunardi, M. Geissert, <a href="http://www.math.unipr.it/~lunardi/Files/OUasympt.pdf" rel="nofollow">Asymptotic behavior and hypercontractivity in nonautonomous Ornstein-Uhlenbeck equations</a>. J. Lond. Math. Soc. (2) 79 (2009), no. 1, 85--106.</p> http://mathoverflow.net/questions/117415/old-books-still-used/117421#117421 Answer by András Bátkai for Old books still used András Bátkai 2012-12-28T17:09:30Z 2012-12-28T17:29:01Z <p>Sz. Nagy-Foias: <a href="http://books.google.hu/books/about/Harmonic_analysis_of_operators_on_Hilber.html?id=ZmpYAAAAYAAJ&amp;redir_esc=y" rel="nofollow">Harmonic Analysis of Operators in Hilbert Space</a> (1970) is a still widely used and lively book (though there is a new updated edition in 2012).</p> <p>T. Kato's <a href="http://books.google.hu/books/about/Perturbation_theory_for_linear_operators.html?id=IvVQAAAAMAAJ&amp;redir_esc=y" rel="nofollow">Perturbation Theory book</a> (1967) is also definitely in this category, though there is a 1980 second edition and a 1995 reprint.</p> <p>Nelson Dunford, Jacob T. Schwartz: <a href="http://books.google.hu/books?id=4LQdAQAAMAAJ" rel="nofollow">Linear Operators</a> (1958,1963, 1971). I still take this book regularly into my hands. </p> <p>An other reference on differential equations is</p> <p>J. L. Lions, E. Magenes: <a href="http://books.google.hu/books/about/Non_Homogeneous_Boundary_Value_Problems.html?id=n7NTQwAACAAJ&amp;redir_esc=y" rel="nofollow">Non-Homogeneous Boundary Value Problems</a>, 1972. It is still "the" reference.</p> http://mathoverflow.net/questions/116150/triangularizing-a-function-matrix-with-smooth-eigenvlaues/116151#116151 Answer by András Bátkai for Triangularizing a function matrix with smooth eigenvlaues András Bátkai 2012-12-12T06:56:44Z 2012-12-12T06:56:44Z <p>This is not a complete answer, but the paper </p> <p>Kreiss, H. O., Über Matrizen die beschränkte Halbgruppen erzeugen, Math. Scand. 7(1959), 71-81.</p> <p>contains a lot of results in this direction. Unfortunately, at the moment I do not have access to it to check...</p> http://mathoverflow.net/questions/114959/counterpart-of-weierstrass-theorem/114960#114960 Answer by András Bátkai for Counterpart of Weierstrass theorem András Bátkai 2012-11-30T07:39:49Z 2012-11-30T07:39:49Z <p>You look for the notion of a <a href="http://en.wikipedia.org/wiki/Pseudocompact_space" rel="nofollow">pseudocompact space</a>. </p> http://mathoverflow.net/questions/110621/easy-question-on-sobolev-spaces/110631#110631 Answer by András Bátkai for Easy question on Sobolev spaces András Bátkai 2012-10-25T08:34:26Z 2012-10-25T08:34:26Z <p>I would like to expand a bit what Delio said. Your question is a bit confusing as it is, but we may assume that you mean that $p$ and $q$ are the parameters representing the derivatives. Then what you need is $$\|f\|_p \leq C\|f\|_q.$$</p> <p>If you write out the definition of the Sobolev norm on $S$, then you see immediately that this holds. This implies that the identity map defined on $S$ extends uniquely and continuously to the whole $H^q$. This gives you the embedding. (Strictly speaking you have to verify that this extension is the identity map, but this follows easily from continuity considerations.)</p> http://mathoverflow.net/questions/109410/nonlinear-delay-differential-equation/109412#109412 Answer by András Bátkai for nonlinear delay differential equation András Bátkai 2012-10-11T20:24:24Z 2012-10-13T17:04:29Z <p>As mentioned already by Denis Serre, there is a rich literature investigating delay equations.</p> <p>If you make an experiment, and fix $\bar{x}=1$, then you see that you need as an initial value the complete past on $[-1,0]$. To play a bit, tak as an initial function the constant function $y(s)=1$ for $s\in[-1,0]$. Then you can calculate the solution explicitly for $x\in[0,1]$, then using this you can calculate the solutuion on $[1,2]$, etc. We see that it is far from being analytic. Hence, no chance for a series sepresentation of a solution.</p> <p>If you are interested in classical stuff, then <a href="http://books.google.hu/books/about/Differential_difference_equations.html?id=j4LwpwGPXzwC&amp;redir_esc=y" rel="nofollow">Bellman and Cooke</a> is an excellent book. An other good reference is the one by <a href="http://books.google.hu/books?id=ZNLjAJQMhqwC&amp;lpg=PP1&amp;hl=de&amp;pg=PP1#v=onepage&amp;q&amp;f=false" rel="nofollow">Hale and Verduyn Lunel</a>.</p> <p><strong>ADDED:</strong> If it is a delay equation (i.e., $\bar{x}>0$), then the initial condition has to be a function (you have to know the whole past). Then the iteration procedure I described works always. This gives you a possible approximation formula, most numerical methods also work this way.</p> <p>You are right about analyticity: series representation does it. Smoothness is a consequence. The example I presented to you is only once differentiable at $x=1$, twice at $x=2$, etc. Hence, cannot be analytic.</p> http://mathoverflow.net/questions/109353/boundary-condition-for-a-non-linear-schrodinger-equation/109361#109361 Answer by András Bátkai for Boundary condition for a non linear schrodinger equation András Bátkai 2012-10-11T07:43:33Z 2012-10-11T07:43:33Z <p>Well, this does not really answer your question, but formally, you cannot define the <a href="http://en.wikipedia.org/wiki/Sobolev_space#Traces" rel="nofollow">trace</a> for functions having regularity $s&lt;1/2$.</p> <p>Practically, it means that if you take fractional power of the Dirichlet-Laplace operator of order less than $1/4$, then boundary conditions disappear from the domain. </p> http://mathoverflow.net/questions/109339/commuting-linear-operators-in-hilbert-spaces/109360#109360 Answer by András Bátkai for Commuting Linear Operators In Hilbert Spaces András Bátkai 2012-10-11T07:30:05Z 2012-10-11T07:30:05Z <p>I am far from being an expert, but there is a list of results for special cases in a book by <a href="http://books.google.hu/books?id=ru3eM5LZgYEC&amp;lpg=PP1&amp;hl=de&amp;pg=PP1#v=onepage&amp;q&amp;f=false" rel="nofollow">Radjavi and Rosenthal</a>, especially in Chapter 9. </p> http://mathoverflow.net/questions/109319/ordered-exponential-of-unbounded-operators/109324#109324 Answer by András Bátkai for ordered exponential of unbounded operators András Bátkai 2012-10-10T19:32:54Z 2012-10-10T20:15:54Z <p>This is usually called the magnus expansion method and has a nice literature in numerical analysis. Kato also used this method to show the existence of solutions in the hyperbolic case. </p> <p>I would say that strong resolvent continuity and a sufficiently big common domain is sufficient in your case. See Section 5.3 in <a href="http://books.google.hu/books/about/Semigroups_of_Linear_Operators_and_Appli.html?id=sIAyOgM4R3kC&amp;redir_esc=y" rel="nofollow">Pazy</a>.</p> <p>I can also give a related <a href="http://arxiv.org/abs/1105.6372" rel="nofollow">self-reference</a>, where also the investigation of this product appears and some of the ideas are explained in a simpler situation.</p> <p><strong>ADDED:</strong> My answer concentrates on the method you propose to converge to the solution. To make the content of the references short: yes. A common dense domain and continuity of hte maps $t\mapsto A_tx$ implies the convergence of the product to the solution of the differential equation.</p> http://mathoverflow.net/questions/109042/reference-request-parabolic-pde/109046#109046 Answer by András Bátkai for Reference request: parabolic PDE András Bátkai 2012-10-07T08:02:41Z 2012-10-07T21:18:53Z <p>If you are interested in sup-norm and Hölder estimates, then Lunardi's book is a good start:</p> <p><a href="http://www.amazon.com/Semigroups-Regularity-Parabolic-Differential-Applications/dp/3764351721/ref=la_B001K6J69O_1_1?ie=UTF8&amp;qid=1349596684&amp;sr=1-1" rel="nofollow">http://www.amazon.com/Semigroups-Regularity-Parabolic-Differential-Applications/dp/3764351721/ref=la_B001K6J69O_1_1?ie=UTF8&amp;qid=1349596684&amp;sr=1-1</a></p> <p>Otherwise you should specify what type of equation are you interested in.</p> <p><strong>ADDED:</strong> Afret the comment of @Liviu:</p> <p>You should not omit Krylov's books from your list: <a href="http://ams.org/bookstore-getitem/item=GSM-12" rel="nofollow">the one on Hölder spaces</a> and <a href="http://ams.org/bookstore-getitem/item=GSM-96" rel="nofollow">the one on $L^p$ spaces</a>. </p> <p>And of course there is <a href="http://www.amazon.com/Partial-Differential-Equations-Graduate-Mathematics/dp/0821849743" rel="nofollow">Evans</a>. An excellent introduction.</p> <p><strong>ADDED:</strong> After the clarification in the question: Topping's <a href="http://homepages.warwick.ac.uk/~maseq/topping_RF_mar06.pdf" rel="nofollow">lecture notes</a> (there is also a book version from the London Mathematical Society) are quite nice and readable.</p> http://mathoverflow.net/questions/37540/reference-for-weak-semigroup/109075#109075 Answer by András Bátkai for reference for weak*-semigroup András Bátkai 2012-10-07T16:11:13Z 2012-10-07T16:11:13Z <p>Echoing the remark of @Bill Johnson, one possibility is <a href="http://books.google.hu/books/about/The_adjoint_of_a_semigroup_of_linear_ope.html?id=9jbvAAAAMAAJ&amp;redir_esc=y" rel="nofollow">van Neerven's book</a> on adjoint semigroups. </p> http://mathoverflow.net/questions/109051/reference-request-schauder-theory-for-fourth-order-parabolic-equations/109072#109072 Answer by András Bátkai for Reference Request: Schauder theory for fourth-order parabolic equations András Bátkai 2012-10-07T15:58:31Z 2012-10-07T15:58:31Z <p>I do not have access to the book you cite, but Section 3.2 in <a href="http://books.google.hu/books?id=mWojiHzg9bEC&amp;printsec=frontcover" rel="nofollow">Lunardi's book</a> contains a broad overview on higher order parabolic problems with lots of references, including the most important Hölder estimates.</p> http://mathoverflow.net/questions/95826/must-neuman-elliptic-operator-has-discrete-spectrum/95873#95873 Answer by András Bátkai for Must Neuman Elliptic operator has discrete spectrum ? András Bátkai 2012-05-03T14:59:20Z 2012-05-03T14:59:20Z <p>Of course, on a general domain, the question os how do you define the Neuman Laplacian. There is an excellent exposition in </p> <p>W. Arendt, A.F.M. ter Elst: <a href="http://arxiv.org/abs/0812.3944" rel="nofollow">Sectorial forms and degenerate differential operators</a></p> <p>suggesting methods how to do it.</p> http://mathoverflow.net/questions/95334/the-exponent-of-self-adjoint-operator/95335#95335 Answer by András Bátkai for The exponent of self-adjoint operator András Bátkai 2012-04-27T09:27:23Z 2012-04-27T09:34:53Z <p>Yes. This follows immediately from the <a href="http://en.wikipedia.org/wiki/Spectral_theorem" rel="nofollow">spectral theorem</a> and form the <a href="http://en.wikipedia.org/wiki/Self-adjoint_operator#Borel_functional_calculus" rel="nofollow">functional calculus</a> of selfadjoint operators, even for a much wider range of functions.</p> http://mathoverflow.net/questions/90129/orthogonality-in-non-inner-product-spaces/90136#90136 Answer by András Bátkai for Orthogonality in non-inner product spaces András Bátkai 2012-03-03T19:24:58Z 2012-03-03T19:24:58Z <p>Well, it depends what do you need it for. You may also have a look at <a href="http://en.wikipedia.org/wiki/Semi-inner-product" rel="nofollow">semi-inner-product spaces</a>, which are natural generalizations of inner product spaces.</p> http://mathoverflow.net/questions/88849/analytic-generator/88862#88862 Answer by András Bátkai for Analytic generator András Bátkai 2012-02-18T22:45:21Z 2012-02-19T20:58:45Z <p><a href="http://books.google.hu/books?id=sOnRZFgR374C&amp;lpg=PP1&amp;hl=de&amp;pg=PP1#v=onepage&amp;q&amp;f=false" rel="nofollow">Arendt-Batty-Hieber-Neubrander</a>: Vector valued Laplace transforms and Cauchy Problems, First Edition, Examle 3.7.6. The Gaussian semigroup. The proof is the same using Fourier multipliers using the excplicite convolution form of the semigroup.</p> <p>Where you find a difference is the question whether the domain of the Laplace is a classical function space. This is only true if $p>1$, then you get a Sobolev space. For $p=1$ the domain is strictly bigger than $W^{2,1}(\mathbb{R}^n)$ for $n>1$.</p> http://mathoverflow.net/questions/87486/reference-request-simple-facts-about-vector-valued-sobolev-space/87615#87615 Answer by András Bátkai for Reference request: Simple facts about vector-valued Sobolev space András Bátkai 2012-02-05T20:53:21Z 2012-02-05T20:53:21Z <p><a href="http://www.amazon.com/Linear-Quasilinear-Parabolic-Problems-Mathematics/dp/3764351144/ref=sr_1_4?s=books&amp;ie=UTF8&amp;qid=1328475127&amp;sr=1-4" rel="nofollow">Herbert Ammann's book</a> on parabolic problems contains an excellent introduction.</p> http://mathoverflow.net/questions/65317/reference-for-neumann-laplacian/78495#78495 Answer by András Bátkai for Reference for Neumann-Laplacian András Bátkai 2011-10-18T22:13:58Z 2011-10-18T22:13:58Z <p>Have a look at Lunardi's book, it is more on the functional analytic questions you have:</p> <p><a href="http://books.google.co.uk/books/about/Analytic_semigroups_and_optimal_regulari.html?id=mWojiHzg9bEC" rel="nofollow">http://books.google.co.uk/books/about/Analytic_semigroups_and_optimal_regulari.html?id=mWojiHzg9bEC</a></p> http://mathoverflow.net/questions/76791/quanitative-de-moivrelaplace-theorem-reference-request Quanitative de Moivre–Laplace theorem (reference request) András Bátkai 2011-09-29T19:46:24Z 2011-09-30T06:20:16Z <p>The classical <a href="http://en.wikipedia.org/wiki/De_Moivre-Laplace_theorem" rel="nofollow">de Moivre-Laplace theorem</a> states that we can approximate the normal distribution by discrete binomial distribution: </p> <p>$${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}e^{-(k-np)^2 / (2npq)}.$$</p> <p>My question is: are there more precise, quantitative versions of this theorem in the literature? Are there good estimates how to measure the error? I am unfortunately not familiar with the subject but need a result of this type. </p> <p>Of course there is always the option of going through existing proofs and checking the details, and turning them from "soft" to "hard", but I suspect this has to be already done. And maybe this is not optimal, maybe there are good accessible ways. </p> <p>Can someone point me a good reference in this direction?</p> http://mathoverflow.net/questions/131413/in-what-rigorous-sense-are-sperners-lemma-and-the-brouwer-fixed-point-theorem-eq/131486#131486 Comment by András Bátkai András Bátkai 2013-05-22T20:56:07Z 2013-05-22T20:56:07Z @James Propp: <a href="http://meta.mathoverflow.net/discussion/1296/2/crank-post-to-flag-as-spam/#Item_9" rel="nofollow">meta.mathoverflow.net/discussion/1296/2/&hellip;</a> http://mathoverflow.net/questions/130857/proof-that-l20-tx-l20-tx Comment by András Bátkai András Bátkai 2013-05-20T19:47:00Z 2013-05-20T19:47:00Z Sorry, I did not get this comment somehow... Yes, it seems to me as easy as you write. http://mathoverflow.net/questions/131203/dual-space-of-bochner-space-is-there-an-easier-proof-to-show-theyre-isometric Comment by András Bátkai András Bátkai 2013-05-20T19:45:17Z 2013-05-20T19:45:17Z See also <a href="http://mathoverflow.net/questions/130857/proof-that-l20-tx-l20-tx" rel="nofollow" title="proof that l20 tx l20 tx">mathoverflow.net/questions/130857/&hellip;</a> http://mathoverflow.net/questions/131070/algorithm-to-find-exponential-map-of-differential-operators-acting-on-function Comment by András Bátkai András Bátkai 2013-05-19T17:49:09Z 2013-05-19T17:49:09Z The formula for the action on $f(x,y)$ is given in detail in the Engel-Nagel reference below. Does it help? http://mathoverflow.net/questions/131037/strong-convergence-in-the-bochner-space-lp0-t-x Comment by András Bátkai András Bátkai 2013-05-18T12:33:06Z 2013-05-18T12:33:06Z Dear Rafa, it seems that some of your formulae is incomplete, something is missing.And, in particular, I miss your question... http://mathoverflow.net/questions/130945/c-c-infty0-tv-is-dense-in-c-c10-tv Comment by András Bátkai András Bátkai 2013-05-17T22:17:19Z 2013-05-17T22:17:19Z A related question: <a href="http://mathoverflow.net/questions/130276/mathcald0-tv-is-dense-in-w0-t" rel="nofollow" title="mathcald0 tv is dense in w0 t">mathoverflow.net/questions/130276/&hellip;</a> http://mathoverflow.net/questions/130972/use-mathematical-induction-to-prove-the-equality-thank-you Comment by András Bátkai András Bátkai 2013-05-17T19:44:12Z 2013-05-17T19:44:12Z As mentioned by Lee, <a href="http://math.stackexchange.com/" rel="nofollow">math.stackexchange.com</a> is a perfect place. http://mathoverflow.net/questions/130936/how-to-proof-this-stirling-related-equation Comment by András Bátkai András Bátkai 2013-05-17T12:46:51Z 2013-05-17T12:46:51Z I am a bit slow, why is the left hand side infinite? http://mathoverflow.net/questions/130857/proof-that-l20-tx-l20-tx Comment by András Bátkai András Bátkai 2013-05-16T19:09:03Z 2013-05-16T19:09:03Z I am a bit confused. If $X$ is a Hilbert space, then $L^2(0,T;X)$ is a Hilbert space (complete + norm comes from a scalar product). Hence if you also identify $X$ with its dual (as you do with $L^2$), then the statement follows. Maybe this is not what you ask? http://mathoverflow.net/questions/130574/why-do-i-get-estimated-error-1-ind-when-doing-bicgstab-linear-solver-using-il Comment by András Bátkai András Bátkai 2013-05-14T12:49:12Z 2013-05-14T12:49:12Z Have you tried to ask it here: <a href="http://scicomp.stackexchange.com/" rel="nofollow">scicomp.stackexchange.com</a> ? http://mathoverflow.net/questions/130354/variation-on-fatous-lemma-for-sobolev-norms Comment by András Bátkai András Bátkai 2013-05-11T21:25:04Z 2013-05-11T21:25:04Z Your point 2 is not the same as above: it follows immediately from the continuity of the norm. http://mathoverflow.net/questions/130368/continuty-of-volume-of-a-convex-set-in-rn Comment by András Bátkai András Bátkai 2013-05-11T21:23:00Z 2013-05-11T21:23:00Z Though I believe this question is better suited at <a href="http://math.stackexchange.com/" rel="nofollow">math.stackexchange.com</a> , let me give you a hint. How do you define your metrics on compact sets? http://mathoverflow.net/questions/130276/mathcald0-tv-is-dense-in-w0-t Comment by András Bátkai András Bátkai 2013-05-10T17:57:21Z 2013-05-10T17:57:21Z Volume 1: <a href="http://rd.springer.com/book/10.1007/978-3-642-65161-8/page/1" rel="nofollow">rd.springer.com/book/10.1007/978-3-642-65161-8/&hellip;</a> , but there are three. http://mathoverflow.net/questions/130276/mathcald0-tv-is-dense-in-w0-t Comment by András Bátkai András Bátkai 2013-05-10T17:55:02Z 2013-05-10T17:55:02Z A standard reference on this is the monograph by Lions and Magenes: Non-Homogeneous Boundary Value Problems and Applications. Everyone refers to it for the proof... http://mathoverflow.net/questions/130242/integrating-a-weak-derivative Comment by András Bátkai András Bátkai 2013-05-10T10:18:47Z 2013-05-10T10:18:47Z Corollary 2.2 here: <a href="http://www.math.psu.edu/bressan/PSPDF/sobolev-notes.pdf" rel="nofollow">math.psu.edu/bressan/PSPDF/sobolev-notes.pdf</a>