Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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27 views

Complexity of a Borel linear subspace of a Banach space

This question is inspired by the MO question Image of $L^1$ under the Fourier transform, but I think it might be much easier so I am posting it here instead. Let $(X, \|\cdot\|)$ be a separable ...
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16 views

Finding the continuity of the mapping of a solution to a PDE to its partial derivative

Here is a modified version of the Black-Scholes PDE: $\frac{\partial \phi(t,S,i)}{\partial t}$ + $r_iS\frac{\partial \phi(t,S,i)}{\partial S}$ + $\frac{1}{2} \sigma^2_i S^2 \frac{\partial^2 ...
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47 views

Compactness hypothesis in Riesz representation theorem

Let $X$ a compact metric space; I have to identify the dual of the set of continuous functions on $X$, $C(X)^*$. By Riesz representation theorem we have that it can be identified with the space of ...
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35 views

Dual space of $L^2(0,T;H^1) + L^p(0,T;L^p)$ and its duality pairing?

Let $V=L^2(0,T;H^1) + L^p(0,T;L^p)$. We know that its dual space is $V^* = L^2(0,T;H^{-1}) \cap L^p(0,T;L^p)$. So if $v \in V$, then by definition $v=a+b$ where $a \in L^2(0,T;H^1)$ and $b \in ...
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110 views

Adjoint of sum = sum of adjoints

is $\mathcal{D}(A)=\mathcal{D}(B)$ a sufficient condition for $(A+B)^*=A^*+B^*$ , where $A$ and $B$ are densely defined (not necessarily symmetric) operators on some Hilbert space?
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35 views

Characterizing direct sums

Let $U,V$ be vector spaces. Let $T: U \to V$ be a linear map. The codimension of $T$ is defined to be $\mathrm{dim}(V) - \mathrm{dim}(\mathrm{im}(T))$. My questions are: (1) given the subspace ...
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30 views

Question concerning continuity of some linear map

So here is my question, I wanted to prove that the canonical embedding of $\ell^p(\mathbb N,\mathbb C)$ in $c_0:=\{(x_n)_{n\in\mathbb N}\subseteq\mathbb C:\lim_{n\rightarrow\infty}x_n=0\}$ is ...
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42 views

Convergence of product of continuous functions and test functions

I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...
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82 views

Hardy-Littlewood maximal theorem (Marcinkiewicz)

I have two pages from a book called "Garnett" and I will present Hardy-Littlewood maximal theorem in class on Wednessday. The theorem is stated: if $f\in L^p(\mathbb{R}), 1 \leq p \leq \infty,$ then ...
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32 views

A problem about the existence of some function in a functional space

Suppose $x\in X\subset \mathbb{R}$. Let $g_{i}(x)$ ($i=\{1,2,\dots,n\}$) be a set of functions in $L^{1}(X,P)$, where $P$ is the probability measure (atomless). Suppose that the functional $$\int ...
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52 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
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49 views

Let $T$ be a bounded operator such that $<Tf,f>=0$ then $T=0?$

Let $H$ be a hilbert space. Let $T:H\to H$ be a linear bounded operator such that $<Tf,f>=0$ for all $f\in H$. It is necesarily true that $Tf=0 ?$ When I mean Hilbert space over a field ...
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64 views

Question about compact operators

I would like to prove the following, Let $X$,$Y$ be infinite dimensional Banach-Spaces and $T$ a compact, linear and bounded operator. Then there exists a sequence $(x_n)_{n\in\mathbb N}$ with ...
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30 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
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53 views

eignvalues of Laplacian operator and distributions

Let $\Omega$ be open and bounded in $\mathbb{R}^n$ and $I$ an interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigenvalues ...
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35 views

Describe all of the functions measurable w.r.t. $\mathcal{A}$ where $\mathcal{A}=\{E \subset X | E \space or \space E^c \space countable\}$

Let $(X,\mathcal{A},\mu)$ be a measure space where $X$ is uncountable, $\mathcal{A}=\{E \subset X | E \space or \space E^c \space countable\}$ and $\mu(E) = \left\{ \begin{array}{lr} ...
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30 views

Density of $C^0([0,T]\times M)$ in $L^p([0,T]\times M)$?

Let $M$ be a compact $C^2$ hypersurface embedded in $\mathbb{R}^n$ of dimension $n-1$. Is the space $C^0([0,T]\times M)$ dense in $L^p([0,T]\times M)$? How do I prove this or what theorem can I use? ...
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72 views

Open ball in infinite dimensional Banach space is not weakly open

I have to prove that open ball in infinite dimensional Banach space is not weakly open. I have no idea how can I do it. I think that I should reach contradiction with infinite dimensions.
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20 views

Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
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31 views

How to prove these equivalences?

I want to prove the following statement: Let $K$ be a compact Hausdorff space and $F\subset C(K)$. Then the following are equivalent: The closure of $F$ in the weak topology of $C(K)$ is weakly ...
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25 views

symmetric quasi-uniformity

A quasi-uniformity $U$ will be called symmetric provided that $U = U^{-1}$, that is, provided that it is a uniformity. Otherwise it will be called nonsymmetric. It is readily seen that the supremum ...
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58 views

An “academic” question on integral operators

This question is motivated by another one, asked by Cameron Williams: Adjoint of an integral operator Let us say that a Borel function $k:\mathbb R\times \mathbb R\to\mathbb C$ defines an operator on ...
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45 views

PDE-Based Triangle Inequality for Optimal Transportation

Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and ...
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52 views

Measure theory integration question involving continuous function

Quick measure theory question. Given that $\Omega \subset \mathbb{R}^{n}$ and $f$ is continuous on $\Omega$. How would you show that if $$\int_{\Omega}f \, dx = 0$$ Then $f = 0$ everywhere? Thanks ...
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69 views

Proving that there is no norm for the space of real-valued sequences making it a complete metric space.

Suppose I have a vector space $K$ which consists of real-valued sequences with only finitely many non-zero terms. I would like to show that there doesn't exist a norm on $K$ that would make it become ...
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52 views

Set of diffeomorphisms on a manifold

It is well known that given a compact smooth boundaryless manifold $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r \geq 1$, is open in $C^{r}(M)$, the set of continuous functions (for ...
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13 views

Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense for $M$ a compact Riemannian manifold?

Let $M$ be a compact Riemannian manifold. Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense?
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40 views

$x_n \rightarrow 0\ (n\rightarrow \infty)$ is stable under a change of topologies

There is an example in the lecture notes I'm currently reading, in a chapter on the dual pairing of a topology, that in $E:=\ell^2$ the convergence of a sequence $(x_n)_n$ to zero in the ...
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24 views

Verification of a contraction

Let $A\colon \text{dom}(A) \to \mathcal{H}$ be a densely defined symmetric operator on a Hilbert space $\mathcal H$. The symmetry implies that $$ \|(A + i)f\|^2 = \|Af\|^2 + \|f\|^2 \quad \text{for ...
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23 views

Existence of the solution of a linear parabolic pde

Good day! Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$. Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in ...
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26 views

Embedding to $L^\alpha(0,T;L^\beta(\Omega))$

Good day! Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$. Consider the space $W = \{ y \in L^2(0,T;V) \colon dy/dt \in L^2(0,T;V') \}$. It is well-known that $W \subset C([0,T];H)$ where $H = ...
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34 views

Lipschitz function and uniform boundedness principle

Let $(S,d)$ be a metric space and $X$ be a normed space. Show that if $f:S\to X$ is a function such that for all $L\in X^*$, $LOf:S\to {\Bbb F}$ is lipschitz(there is a constant $M>0$ such that for ...
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31 views

Properties of this set of functionals (mixed pairings)

(from the 4th page of http://www.math.toronto.edu/mccann/papers/econ.pdf) Let $X$ be a compact Hausdorff space, and let $\omega$ be a Borel probability measure on $X$. A Borel probability measure, ...
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34 views

On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
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53 views

Continuity of the identity operator from weak to weak star topology

I have a question in functional analysis. Suppose $X$ is a normed space and $I:X'\rightarrow X'$ is an identity operator such that $If=f$ for $f$ in $X'$. ($X'$ is the dual space of $X$). Show that ...
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56 views

Is a core for the generator of a Feller semi-group invariant under the resolvent?

Let $\{T_t:t\geq 0\}$ be a Feller semi-group acting on $C_0(\mathbb{R})$ with generator $(A,\mathcal{D}_A)$. We know a subspace $D\subset \mathcal{D}_A$ is a core for $A$ if $(\lambda-A)D$ is dense in ...
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62 views

Separability of nuclear spaces

I'm currently studying nuclear spaces and the nuclear spectral theorem with the books of Gelfand, Vilekin and Schilow, "Generalized functions", especially the second and fourth part. Currently, I'm ...
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30 views

About a vectorial measure

We consider the measure space $\mathbb{N}$, with the counting measure, and $\lbrace{e_i\rbrace}$ the standard basis for $\ell^1$: How can I prove that the function $\nu$ mapping each ...
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63 views

Bounded Sesquilinear form

Let $X$ and $Y$ be normed spaces. Show that a bounded sesquilinear form $h$ on $X \times Y$ is jointly continuous in both variables.
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37 views

Want to prove an inequality of two norms in a Hilbert space

So here is my problem, Let $D:=[-d,d]\times[-d,d]$ and $C_0^{\infty}$(D) be the set of all smooth functions with compact support in $D$ which are zero on the boundary of $D$. Moreover we have the ...
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49 views

the principle of uniform boundedness

If $\{x_n\} \subset \ell^1$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in c_0$ iff $\sup_n||x_n||_1< \infty$ and $x_n(j)\to 0$ for $j\geq 1$. I can proof it by the principle of uniform ...
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26 views

Limit of function of an operator

Let $A_n$ be a sequence of bounded, self-adjoint operators on Hilbert space $\mathcal{H}$. Let us assume that for some vector $\psi\in\mathcal{H}$, $$\lim_{n\rightarrow\infty}A_n\psi = \alpha ...
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60 views

If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research: Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
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30 views

Question about Green function

how to find the Green function of this problem : $$ \begin{cases} -(p(t)u'(t))'=\lambda f(t,u(t)) ~ \text{a.e.} ~t>0\\ u(0)=u(+\infty)=0 \end{cases} $$ Thank you
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50 views

bilinear form, anti symmetric part

$\mathcal{H}$ : real Hilbert space with inner product $(\,,\,)$ and norm $||\,||:=(\,,\,)^{1/2}$ Let $D$ be a linear subspace of $\,\mathcal{H}$ and $\mathcal{E}$ : $D\times D\to \mathbb{R}$ a ...
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78 views

When do partial subgradients give a subgradient?

I'm looking for sufficient conditions that guarantee that partial subgradients of a convex, lower-semicontinuous functional $f:X_1\times X_2\rightarrow\overline{\mathbb{R}}$ form a subgradient of $f$. ...
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34 views

Spectrum of the operator of differentiation along streamlines

Suppose $\mathbb T^2$ denotes the two-torus and suppose $\psi^0$ is a steady state smooth stream function on $\mathbb T^2$ and define $\mathbf u^0 = (-\partial_y \psi^0,\partial_x \psi^0 )$ be the ...
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25 views

Simultaneous extension and complemented subspace

The following is Exercise 3.13.5 of Conway's Functional Analysis: Let $X$ be a compact set and let $Y$ be a closed subset of $X$. A simultaneous extension for $Y$ is a bounded linear map $T:C(Y)\to ...
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35 views

Linear map in Hilbert space.

If you have a linear map $h\mapsto T(h)$ from $H_1$ a real separable space, to Hilbert space $H_2$, it seem that this maps provides an isometry of $H_1$ onto a closed subspace of $H_2$. I try to ...
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67 views

$C^1[0,1]$ is not complete with respect to sup norm

I think the right sequence is $f_n(t)=|t-\frac{1}{2}|^{(n+1)/n}$ but I can't manage to prove it's Cauchy... (when I look at graph of $f_n$s it seems obvious, but I need it formal way.