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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51 views

ODE with periodic initial/end condition

Are there any existence/uniqueness results for solutions to the ODE $$y'(t) = f(y(t),t)$$ $$y(0) = y(T) = y_0$$ on the time interval $[0,T]$ where $f$ is Caretheodory and $y_0$ is given. I am looking ...
4
votes
1answer
121 views

Measurable structure on the space of probability measures

My advisor only half-jokingly mentioned that sometimes people like to consider the measurable structure on $P(X)$ where X is a locally compact polish space and $P(.)$ denotes the probability measures ...
3
votes
1answer
188 views

Closed-form expressions for dual norms of real normed vector spaces

Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\|v\|$. Then say that $V^*$ is the dual space of linear functionals on $V$. The "dual ...
2
votes
1answer
105 views

A Banach space with multiple preduals

Is it possible to have two (separable) Banach spaces, $X$ and $Y$, that are not isometrically isomorphic, and yet their dual spaces $X^*$ and $Y^*$ are isometrically isomorphic?
3
votes
3answers
67 views

Proving $\|T^n\|\leq \|T\|^n$

Let $T:X\to X$ be a linear bounded mapping. I have to prove $\|T^n\|\leq \|T\|^n$. Let $Tx=cx$, where $c>0$. This is a linear mapping. $$T^2 x=T(Tx)=T(cx)=cTx=c^2 x.$$ Hence $\|T^2x\|=c^2\|x\|.$ ...
4
votes
1answer
245 views

Existence of a Lagrange multiplier (Euler Lagrange equations + holonomic constraints )

Let $I=[a,b]\subset \mathbb{R}, G:\mathbb{R}^n\to \mathbb{R}^k$ smooth, $0<k<n, M=G^{-1}(0)$. Assume that $DG(x)$ has full rank for all $x\in M$. Fix $p_1,p_2\in M$ and assume $u\in ...
4
votes
1answer
90 views

How do I derive p-Laplacian?

How do I obtain the p-Laplacian equation $$\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$$ as the minimiser of the integral $$\int_{\Omega}|\nabla u|^p$$ ? I can't expand $|\nabla u + ...
6
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2answers
717 views

Does $\mathcal{L}^2(\mathbb{R})$ form a metric space with this distance/similarity measure?

Consider the set $\mathcal{L}^2(\mathbb{R})$, where two functions $f$ and $g$ are said to be equal, if they agree almost everywhere. I would like to define a distance/similarity measure and would like ...
0
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1answer
102 views

Holder's and Minkowski's Inequalities

I can prove the summation aspect of both Holder's and Minkowski's Inequalities but their integral aspect of the proof is difficult for me to prove. Can I get any approach in proving the integral ...
2
votes
1answer
72 views

Show that $\max_i\left\{|v_i| + |w_i|\right\} \leq \max_i\left\{|v_i|\right\} + \max_i\left\{|w_i|\right\}$.

While trying to prove that the $\infty$-norm of a vector in $\mathbb{R}^n$ does satisfy the properties of being a norm, I inevitably came across the following inequality: $\max_i\left\{|v_i| + ...
1
vote
1answer
63 views

existence of a weakly cauchy sequence if the dual space is separable [closed]

Let X be a normed space such that $X^*$ is separable. Given any sequence $(x_n)\in X$ then there exist a subsequence weakly cauchy
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votes
1answer
65 views

$(u_n)$ is bounded in $H^1(\mathbb{R}^N)$, some results about the convergence.

If $(u_n)$ is bounded in the Hilbert space $H^1(\mathbb{R}^N)$, we have that, up to a subsequence, \begin{eqnarray} &&u_n \rightharpoonup u\ \mbox{ weakly in }H^1(\mathbb{R}^N),\\ ...
5
votes
1answer
67 views

Compactness with Ascoli-Arzelà?

Let $$ K:= \{x: [0,T] \to \mathbb R: x^{\prime}(t)=x^2(t), \, 0\le x(T) \le 1\}. $$ Prove that $K$ is a compact subset of $C([0,T],\mathbb R)$. My idea is to use Ascoli-Arzelà thm. First of ...
1
vote
2answers
773 views

Prove that if a linear operator is continuous, then it is bounded.

I'm trying to prove that if a linear operator is continuous, then it is bounded. Let $T:V\to W$. Let us assume it is continuous. Then for any $\epsilon>0$, $\|T(x-x_0)\|<\epsilon$ if ...
0
votes
1answer
32 views

Image of innerproduct unordered field?!

This might be totally stupid, but I have a problem regarding IPS and NLS. My problem is that as I understand, one does not assume that the image of the innerproduct is an ordered field..? In the ...
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0answers
221 views

Convex hull of bounded set is bounded

I want to show that in a locally convex topological vector space $X$, the convex hull of a bounded set is bounded. Apparantly this does not hold if $X$ is not locally convex. So the fact that that ...
5
votes
1answer
272 views

Uniform sampling of points on a simplex

I have this problem: I'm trying to sample the relation $$ \sum_{i=1}^N x_i = 1 $$ in the domain where $x_i>0\ \forall i$. Right now I'm just extracting $N$ random numbers $u_i$ from a uniform ...
3
votes
1answer
53 views

Proving $||A||=||A^{*}||=||AA^{*}||^{1/2}$

I am studying functional analysis, In the lecture notes I saw the claim: Let $A\in L(\mathcal{H})$ where $\mathcal{H}$is a Hilbert space. then $$ ||A||=||A^{*}||=||AA^{*}||^{1/2} $$ There is ...
4
votes
1answer
163 views

A detail in the proof of Banach-Steinhaus theorem that I don't understand

I am studying functional analysis and I have seen the Banach-Steinhaus theorem. For starters, the motivation given was the question about when $\{T_{\alpha}\}_{\alpha\in A}$ are bounded by $M$ (here ...
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2answers
67 views

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$ I need to know whether it is self adjoint and unitary operator given that $x_i\in\mathbb C$ I am not able to do it please tell me how ...
2
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3answers
63 views

$I_{id}:(C[0,1],\|\|_{\infty})\to (C[0,1],\|\|_{1}) $ is continuous and the open map?

$I_{id}:(C[0,1],\|\|_{\infty})\to (C[0,1],\|\|_{1}) $ is continuous and the open map? $\|f\|_{\infty}=\sup_{x\in [0,1]}\{|f(x)|\}\le\int_{0}^{1} |f (x) |$ so it is continuous and homeomorphism so ...
3
votes
2answers
171 views

Is it a unitary, self adjoint and normal operator?

Let $A\colon H\to H$ be a bounded linear operator on a complex Hilbert space such that $\|Ax\|=\|A^*x\|\forall x$, given that there is a nonzero $x$ for which $A^*x=(2+3i)x$. Then I need to know ...
1
vote
0answers
86 views

Show that $(D_t t)$ is an isomorphism.

Let $B$ be a Banach space, $\epsilon > 0$, and $$C_1^0 ([-\epsilon,\epsilon],B) = \{u \in C^0([-\epsilon,\epsilon],B) : tu \in C^1([-\epsilon,\epsilon],B)\}.$$ Denote $\partial/\partial t$ by ...
3
votes
2answers
187 views

Continuity+differentiability imply weak sequential continuity?

Let $E$ and $F$ be two Banach spaces. We know that, if $f:E\rightarrow F$ is a nonlinear continuous operator, then $f$ may fail to send weakly convergent sequences to weakly convergent sequences, ...
2
votes
0answers
72 views

Show the mapping is $C^1$

I have the following problem: Suppose $f\in C^1(\mathbb{R},\mathbb{R})$. Let $C([0, 1])$ be the space of continuous functions with norm $||u||_{\infty} = \max_{x \in [0, 1]} |u(x)|$. Show that the ...
5
votes
0answers
124 views

Importance of Schwartz kernel theorem

I am currently reading the proof of the Schwartz Kernel Theorem from Hormander Vol I. At the risk of sounding naive, what is the importance of Schwartz kernel theorem? What are certain insights that ...
3
votes
2answers
289 views

Dido's problem with Euler equations

I'm considering Dido's problem: Consider 2 differentiable arcs $C$ and $C_0$ in $\mathbb{R}^2$ from the point $P$ to $Q$ and back. We keep $C_0,P,Q$ fixed, and want to choose the arc $C$ such that ...
2
votes
1answer
88 views

Inverse problem in calculus of variations

I am interested in knowing which differential equations follow from a variational principle. I am reading this and it provides the answer for ordinary differential equations. Is there a complete ...
1
vote
1answer
93 views

Isometry from Banach Space to a Normed linear space maps

Let $X$ be a Banach Space and $Y$ be a normed linear space. Show that if $T$ is an isometry then $T(X)$ is closed in $Y$. Let me have some idea to solve this. Thank you for your help.
3
votes
1answer
161 views

Projective limit of Banach spaces

Let $(X_s)_{s \in (0,s_1)}$ be an increasing sequence of Banach spaces with the property that if $0<s<r<s_1$, then $$ \|u\|_{X_s} \leq \|u\|_{X_r}. $$ We define $$\tilde{X}_s = ...
2
votes
1answer
63 views

A question about quotient space of $R(T^{n})$

I am reading a paper about spectral theory. The author says it is easy to see the following proposition: For $T\in L(X)$, if dim$(R(T^{d})/R(T^{d+1}))<\infty$, then $R(T^{d})$ is closed if and ...
3
votes
2answers
79 views

Is the following and isomorphism $H_0^1(\Omega) \to H^{-1}(\Omega)$?

While working with differentiation with respect to the domain I stumbled upon the following question. Let $\Omega \subset \mathbb R^N$ be bounded, $\alpha \in L^\infty (\Omega)$ be non-negative and ...
4
votes
1answer
305 views

Order of distribution

Let $T$ be Schwartz distribution. Assume that the following inequality holds $T(\phi) \leq \textrm{const} ~\| \tilde{\phi}\|_1$ for any $\phi \in S(\mathbb{R})$ ($\tilde{\phi}=\mathcal{F}(\phi)$ is ...
2
votes
2answers
39 views

How can I show $\lim_{t\to 0^{+}}\|S(t)-I\|\neq 0$ where $S(t)f(x)=e^{-t^2-2tx}f(x+t)$..

I need some help with the following: For every $t\in [0, \infty)$ let $S(t):C_0([0, \infty))\rightarrow C_0([0, \infty))$ be the bounded linear operator given by, $$S(t)f(x)=e^{-t^2-2tx}f(x+t)),$$ ...
2
votes
2answers
98 views

Dimension of space of continuous functions

Please can you help me in this exercice. Prove that the normed space of continuous functions $f: [0,t] \to \mathbb{R}$ with the fundamental norm $||\cdot||_2$ is infinite dimensional. ...
1
vote
1answer
142 views

Identity with Dirac delta function: $\delta (x^2-a^2) = \frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]$

How can I show that $\delta (x^2-a^2) = \frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]$? I'm suppose to integrate it by a differentiable function and integrate, but I can't figure this one out.
3
votes
1answer
54 views

What does it mean for a surface to evolve with divergence-free velocity?

Suppose we have an evolving hypersurface which evolves with a velocity field $V$, such that $\nabla_S \cdot V = 0$ where $\nabla_S$ is the surface or tangential gradient. What does this mean? What ...
2
votes
2answers
114 views

problem metric involving sequences

UPDATED: In preparation for an exam I am struggling with the following problem. We have $A:=\{x=(x_{n})_{n}\in \ell^{2}| \phantom{x} \|x\|\leq 1\}$. Consider the metric $d:A\times A \rightarrow ...
4
votes
2answers
225 views

How to show that $p-$Laplacian operator is monotone?

Define $$\langle -\Delta_p u, v \rangle_{(W^{1,p})', W^{1,p}} = \int_{\Omega}|\nabla u |^{p-2}\nabla u \nabla v.$$ How do I show that this operator is monotone? I get $$\langle-\Delta_p u_1 - ...
1
vote
2answers
44 views

how to find the norm here?

$T_n:l_2\to l_2; T_n(x_1,x_2,\dots)=(0,0,\dots,x_{n+1},x_{n+2},\dots)$ could any one help me how to find the norm here? What is the difference between $\|T_n\|\to 0$ and $\|T_n(x)\|\to 0$? Here both ...
1
vote
2answers
58 views

Existence of solution in Hölder spaces

Let's say we have a PDE, for example the Laplace equation: $$ \Delta u = f. $$ Usually, to solve such a thing, one finds its variational formulation, and solves it in some Sobolev space. My question ...
1
vote
1answer
61 views

Are $T,T^2$ compact operators?

$T:l_2\to l_2$ is defined by $T(x_1,x_2,\dots)=(0,x_1,0,x_3,0,x_5,\dots)$ we need to find whether $T, T^2$ are compact or not. I see here the definition of compact operator but I'm not able to apply. ...
1
vote
1answer
46 views

Is this set open in the Euclidean topology on the plane?

Let $\mathbf{R}^2$ be the two-dimensional Euclidean space, and let $$ A := \{ (x,y) \in \mathbf{R}^2 | \, \, \, |x| < \frac{1}{y^2+1} \}.$$ Then how can we establish (preferably using the ...
4
votes
1answer
227 views

The set of all continuous functions on a locally compact Hausdorff space.

I am reading a book about C*-algebra. There is a example that i could not understand. Let $X$ be a locally compact Hausdorff space and $C_{0}(X)$ be the set of all continuous functions vanishing at ...
0
votes
1answer
89 views

A metric inducing the topology of pointwise convergence on a bounded subset of $\ell^2$

We have $A:=\{x=\{x_{n}\}\in \ell^{2}| \phantom{x} \|x\|\leq 1\}$ Consider the metric $d:A\times A \rightarrow [0,\infty)$ defined by $$d(x,y)=\sum_{n=1}^{\infty}(1/2)^{n}|x_{n}-y_{n}|$$ Show ...
3
votes
1answer
139 views

A real analysis question

I would like to ask if the following statement is true or not: Let $u,v:\Omega\subset% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{N}\rightarrow% %TCIMACRO{\U{211d} }% ...
3
votes
2answers
502 views

Prove that a norm that satisfies the parallelogram inequality defines an inner product

In the opening chapter of a functional analysis book I had this question: Prove that for a norm $||\cdot||$ that if for all vectors $u$ and $v$ it is true that $2||u||^2 + 2||v||^2 = ||u+v||^2 + ...
1
vote
1answer
84 views

The operator norm of complex matrices

Let $M_{n}$ be the algebra of $n\times n$ complex matrices. By identifying $M_{n}$ with $B(\mathbb{C^{n}})$, the set of all bounded linear maps from the n-dimensional Hilbert space $\mathbb{C^{n}}$ to ...
2
votes
1answer
65 views

How to show this Sobolev space is a uniformly convex space?

From the book by Kufner: How do I prove this theorem? I'd like to do it using the epsilon delta definition (see http://en.wikipedia.org/wiki/Uniformly_convex_space) if possible.
0
votes
1answer
63 views

Approximating linearly independent functions with linearly independent functions.

Let $(\Omega,\Sigma,\mu)$ be a measure space. Let $f_{1}, ... , f_{n}$ be an $\bf{\text{Auerbach basis}}$ for a finite dimensional subspace $N\subset L_{1} := L_{1}(\Omega,\Sigma,\mu)$. That is, ...