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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33 views

Approximating a $C^1$ function by piecewise affine maps

Let $\Omega\subset\mathbb R^n$ be an open and bounded domain and let $f\in C^1(\bar\Omega,\mathbb R^m)$. I would like to approximate $f$ by a function $u:\bar\Omega\to\mathbb R^m$ that is piecewise ...
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19 views

existence problem

Let $\Omega=\mathbb{R}^2_+ = \{(x,y) \in \mathbb{R}^2; y>0\}$, $\Gamma=\{(x,0), x\in \mathbb{R}\}$. We consider in $\Omega$, the problem $$-\mathrm{div}(A \nabla u) + \lambda u = f, (x,y) \in ...
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26 views

Why do the following regularisations of a function in Sobolev space exist?

Suppose $v_1, v_2$ satisfy $\mu \leq v_1(x,t), v_2(x,t) \leq M$ a.e. in $Q:=\Omega\times(0,T)$ and $$(v_1, \eta_1) \quad\text{and} \quad (v_2, \eta_2) \in L^2(0,T;H^1(\Omega)) \cap L^2(Q).$$ Define ...
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23 views

Show that $\lim_{a \to \infty} \sup_{n} \int_0^{T-a}||v_{n,r}(t+a)-v_{n,r}(t)||_{\mathbb{L}^2(\Omega_{2r})}^2dt=0.$

Let $\ 0 \leq t \leq t+a \leq T$, with $$\lim_{a \to 0} \sup_{n} \int_0^{T-a}\left\|u_n(t+a)-u_n(t)\right\|_{\mathbb{L}^2(\Omega_{r})}^2dt=0,$$ where $\Omega_r=\Omega \cap \left\{x \in \mathbb{R}^2; ...
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62 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
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60 views

Need help with understanding the argument in the proof

I am working with one theorem in the book " Topics in Banach Space Theory" by Fernando Albiac and Nigel J.Kalton. I have put a full proof of the theorem here. Here is the book if you want to have a ...
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41 views

Find all functions such that $\sum_{k=1}^{n}\frac{1}{f(k) \cdot f(k+1)} = \frac{f(f(n))}{f(n+1)}$

Please help me to solve this problem. Any ideas please comment and share it. Thanks you before hand for helping! :) Determinte all functions $f: \Bbb N \to \Bbb N$ such that ...
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13 views

Operator's comparison

I have found the following two concepts: $\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$ and for any ...
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49 views

Show the following extension is lipschitz

$X=S\cup\{x_0\}$, $f:S\rightarrow \mathbb R$ s.t. $|f(s)-f(t)|\leq kd(s,t)$ for $s,t\in X, k>0$. Suppose $x,y \in X$ s.t. $x\in S$ and $y\notin X$then $x=t, t\in S$ and $y=x_0$. I'm trying to ...
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51 views

Infinite solutions of Navier-Stokes equations

Is it a known fact that Navier-Stokes equations have exactly one (possibly infinite) solution in the space of distributions?
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32 views

Property of the inverse image of rationals under $f(x)=\tan x$

Let $K$ be the set of all $x\in \Bbb R$ such that $\tan x\in \Bbb Q$ or $\pm \infty$. Is there a nonconstant function $m:K\to \Bbb R$ such that $$ m(x+y)=m(x)m(y),\quad m(\pi)=1 $$ for all $x,y\in ...
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27 views

Problem about Projections

can someone help me with the following problem: Let $X$ be a Banach space and $P \in L( X,X)$ be a projection. Show that $P$ is open. where $L(X,X)$ is the space of bounded linear operators from ...
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42 views

Seemingly easy analysis problem but unsure how to proceed.

if $f(x)=\frac{1}{x+2}$ then $f(x)=1-(x+1)+(x+1)^2+T$ for some $x_0$ between $x$ and $-1$ where $T=-\frac{(x+1)^3}{(2+x_0)^4}$ I'm not sure how to proceed in solving this problem. We recently ...
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43 views

When integration by parts applies?

I think integration by parts is really useful, but I don't know when integration by parts applies. Take the following proposition as an example: Let $\Omega \subset \mathbb{R}^n$ be an open set, ...
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135 views

Set of infinitely differentiable functions compactly supported in a domain of $\mathbb{R}^n$ not dense in $L^\infty$

How does one show that the set of infinitely differentiable functions compactly supported in a domain $\Omega\subset\mathbb{R}^n$ is not dense in $L^\infty(\Omega)$? Thanks!
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33 views

Congruences or logs

How do you know the pairs of integers $x,y$ such that $$y=\frac{ln(p(x))}{log(k)}$$ is true, where $p(x)$ is any Diophantic equation and $k$ any Natural number?
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23 views

Derivative of function on parametrized manifold

Given $U\subset\mathbb{R}^d$. Let $\alpha:U\mapsto \mathbb{R}^k$ be an injective function such that $\alpha(U)$ is $d-$dimensional parametrized manifold. Now define $\beta:\alpha(U)\mapsto U$ by ...
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27 views

Question about $Y^{\perp}$

Let $Y$ be a subspace of a normed vector space $X$. Let $Y^{\perp}$ be the space of all $\lambda \in X^{\ast}$ which vanish on $Y$. Why is $Y^{\perp}$ trivial if and only if $Y$ is dense and ...
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24 views

Question on the Deformation lemma

I have a functional $J$ defined on a hilbert space, with a finit number of a critical point $v_1,...,v_m$ let $b>\max\lbrace J(v_1),...,J(v_2)\rbrace$, and i want to prove that the set ...
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85 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
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15 views

Is this map Hilbert Schmidt?

Let $\Sigma$ be a $2$ dimensional closed oriented Riemannian manifold. The Sobolev space $W^1(\Sigma,m)$ is the completion of $C^{\infty}(\Sigma,\mathbb{R})$ with respect to the inner product ...
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48 views

topological vector space of measure functions

Let $(X, \mathcal X, \mu )$ be a measure space, and let $ L(X)$ be the space of measurable functions $f: X \to \mathbb C$. Show that the sets $B(f, \epsilon ,r ): = \{ g \in L(X) : \mu( \{ x : | f(x) ...
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47 views

It is possible to find such functions?

I would like to know if it is possible to find a couple of functions $f,g$ such that $f*g$ and $g*f$ exists and such that $f*g\ne g*f$ ? if not it would mean that the convolution product commutes ...
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35 views

A vector space with topology generated by a family of typologies each makes it a topological vector space is a topological vector space

Let $V$ be a vector space, and let $(\mathcal F_ \alpha ) _{ \alpha \in A}$ be a family of topologies on V, each of which turning $V$ into a topological vector space. Let $\mathcal F$ be the vector ...
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31 views

Proof Check: Closed range then bounded below

Statement: Given a Hilbert $\mathscr{H}$, and $T \in \mathscr{B}(\mathscr{H}, \mathscr{H})$, where $T$ has closed range. Prove that for all $h \in N(T)^\perp$ then $\exists \, m>0 \, \mbox{s.t.} \, ...
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15 views

Epi-convergence to indicator function

Let the indicator function be defined as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and $I_{\nu}(x) \in [0,1]$ be a continuous approximation of ...
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55 views

Show these operators converge to a particular limit

Let $H$ be a Hilbert space, and $T$ be a operator on $H$ of the form $T=\sum_{n=1}^{\infty}{\lambda}_{n}<x,e_{n}>e_{n}$ where $e_{n}$ are the eigenvectors of $T$ and an orthonormal basis of H ...
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26 views

Minimizing the homogenuos Sobolev norm for a given trace

Suppose that $\Omega$ is a bounded domain with regular boundary (think $C^1$). We have a function $f_b:\partial\Omega\to\Bbb R$ and we can expand it to the whole $\Omega$ in the sense of ...
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25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
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30 views

Fourier transform of a function of characteristic function of a measure

Let $\mu$ be complex measure on $\mathbb{R}^2$ ($|\mu|$ is finite measure) and $\chi$ - its characteristic function $$ \chi(x_1,x_2) = \int_{\mathbb{R}^2} d\mu(p_1,p_2) \exp(i p_1 x_1+i p_2 x_2). $$ ...
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28 views

Rotating the spectrum of a bounded operator

If $T$ is a bounded operator on a Banach space $X$, and $\sigma(T)$ is its spectrum, what would be an operator whose spectrum is $\sigma(T)$ rotated by $\theta$? For example, $-T$ has as spectrum ...
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18 views

Differentiation by definition

I want to show that the function $f:\mathbb{C}\rightarrow A$ given by $f(x)=e^{ixb}ae^{-ixb}$ (where $A$ is a unital $C^*$ algebra and $a,b\in A$ fixed) is differentiable with $f'(0)=i(ba-ab)$. My ...
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22 views

Bilinear form on the space of smooth complex valued functions.

Let $G$ be a Lie group and $h$ be the Hermitian bilinear form on smooth complex valued functions then how can we define bilinear form on the space of smooth complex valued functions.
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24 views

Show that $d\log f$ is a 1-current on a 1-dimensional complex manifold

I am having trouble with this problem (and it might be because I have the formulation slightly off). I need to show that $d\log f$ is a 1-current on a 1-dimensional complex manifold $M$. This means ...
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21 views

Linear functional

Let $g:P\rightarrow \Re$ be a sublinear functional, $P\subset X$ is a convex cone and $X$ a normed vector space. Let $c>0$ be any constant such that $g(y)>c$ for any $y\in P.$ Do there exist ...
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28 views

The difference between an operator and a function

In functional analysis we are introduced with a new notion: the notion of operators. It has been mentioned before (linear operators in linear algebra, derivative as an operator and integral as an ...
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26 views

Why Steiner Symmetrization makes a measurable set to a measurable one?

I find the Steiner Symmetrization is very useful in proving that the Hausdorff measure coincide with Lebesgue in the Euclidean space. However, I never saw anybody mention that the Steiner ...
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43 views

continuous functional

Let $E$ be a Banach space and $\Phi(x)$ is a continuous convex functional of $x \in E$. I have found The following inequality $$\Phi(x) \leq\|\Phi\|\|x\|$$ and I haven't understand it and what's the ...
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15 views

Morphism of $G$-modules, representation in banach space

We have some group $G$ and it representation in banach space $X$. It means that there is a group-homomorphism $T:G\longrightarrow \mathrm{GL}(X)\cap \mathfrak{B}(X)$, where $\mathfrak{B}(X)$ is a set ...
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28 views

Besov Spaces References..

Does anyone know any good modern reference about Besov spaces? The only references I found was some Triebel's books as "Interpolation Theory, Function Spaces, Differential Operators" and "Theory of ...
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51 views

Studiy of a differential operator

Let $V=W^{1,p}_0(\Omega)$ and this dual space $V'=W^{-1,p'}(\Omega)$ with $p'$ the conjugate of $p$. Let $A(u)=-\mathrm{div}(|\nabla u|^{p-2} \cdot \nabla u)$. How we can prove that $A$ is monotone? ...
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32 views

operator onto-theorem

I have this theorem: Let $V$ a Banach space, reflexive,separable, and let $A$ an operator monotonic, bounded, semi-continuos, coercive. Then, $A$ is onto. Where we can find the proof of this ...
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28 views

Bounded inverse theorem.

http://planetmath.org/boundedinversetheorem referring to this proof i don't get the final statement: "$T^{-1}$ is continuous, i.e. bounded". I know that the boundness would be surely true if $T^{-1}$ ...
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21 views

$T$ is injective iff $T(H)$ linearly independent.

Let $T:X\to Y$ be a linear transformation and $H$ be a hamel basis of $X$, then prove that, $T$ is injective iff $T(H)$ linearly independent.
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25 views

Why this relation in Hilbert space (with inner product $< >$) holds?

$c_k$, $f_k$ are sequences in Hilbert space, $g$ is a function. Why this relation below holds? How you derive it? $\sum_k|c_k\left \langle f_k,g \right \rangle|\leq(\sum_k|c_k|^2)^{1/2}(\sum_k|\left ...
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24 views

Summability of Fourier series from Banach space point of view

I am under the impression the following is true (any pointer to a reference would be appreciated ): Theorem (Katznelson?) For any $f \in C[0,1]$ with Fourier coefficients $\{ \hat{f}(n)\}$, there ...
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22 views

Regularising a function that is constant on an interval (related to Heaviside)

Define the function $f:\mathbb R \to \mathbb R$ by $$f(x) = \begin{cases} x &\text{for $x < 0$}\\ 0 &\text{for $x \in [0,1]$}\\ x-1 &\text{for $x > 1$}& \end{cases} $$ Note that ...
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32 views

Conditions on $\alpha_n$ for $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ to be a norm on $l_p$

When $n_\alpha(x)=\sum^\infty_{n=1} \alpha_n\vert x_n\vert $ is a norm in $\mathcal{l}_p=\lbrace (x_k)^\infty_1 : \sum\vert x_k\vert ^p \lt\infty\rbrace $ and $\alpha\in\omega$. and $\omega$:space of ...
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36 views

The Kobayashi pseudo - distance $d_X$ and the Carathéodory pseudo - distance $c_X$

I'm studying the Kobayashi pseudo - distance $d_X$ and the Carathéodory pseudo - distance $c_X$. And I have trouble when I try to show $4$ properties of $c_X$ in my textbook. {It doesn't have ...
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21 views

Variational methods

I have this BVP $$\begin{cases} u^{(4)}=f(t,u(t)),\, t\in[0,1]\\u(0)=u(1)=u''(0)=u''(1)=0\end{cases}$$ how to prove that the functional asociated to this BVP is $$J(u)=\frac12 ||u||^2-\int_0^1 ...