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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140 views

Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
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1answer
47 views

Continuous function between measure spaces is measurable

Suppose $$F(t) = \lVert {f(t)} \rVert_{L^2(\Omega_t))} \tag{1}$$ is a continuous as a function of $t$ for each $t \in [0,T].$ A continuous function between measure spaces is measurable, so (1) is ...
1
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1answer
472 views

Show $\mathbb R^n$ is complete.

Show $\mathbb R^n$ is complete. At this point, I am trying to work through the problem in my textbook, there is one step that I do not understand and would like explained. Here's my proof so far: ...
1
vote
1answer
218 views

Lipschitz condition on a second order nonlinear ODE?

Preliminaries: Let the matrix norm be $$\sqrt{\sum_{j=1}^n\sum_{i=1}^n a_{ij}^2}=||\mathbf A||.$$ I am trying to prove uniqueness and existence of a second order nonlinear ODE (Ordinary Differential ...
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1answer
46 views

Show that for all functions f, g … the following equivalence holds

Would you please help me to solve this exercise? Show that for all functions $f,g:\,\mathbb{N\,\rightarrow\mathbb{R_{>0}}}$ the following equivalence holds $$f(n)\in ...
4
votes
1answer
121 views

Why is there no space whose dual is $C_\mathbb{R}[0,1]$? [duplicate]

Possible Duplicate: $C_0(X)$ is not the dual of a complete normed space Is any Banach space a dual space? While studying for a course of functional analysis I read somewhere that there is ...
2
votes
0answers
260 views

Fredholm and Compact Operators

Let $X$ and $Y$ be Banach spaces and $T\in B(X,Y)$ be Fredholm. Then there is $S\in B(Y,X)$ such that $ST=I+K_{1}$ and $TS=I+K_{2}$ where $K_{1},K_{2}$ are compact operators.Proof: Since $T$ is ...
2
votes
1answer
55 views

Is there a natural topology on C(X), if X is infinite-dimensional?

Suppose $X$ is an infinite-dimensional Banach space. Is there a natural topology on $$C(X)=\{f:X\to\mathbb{R}: \text{ $f$ is continuous}\}?$$
6
votes
3answers
199 views

Is a closed $G_\delta$ set in a Hausdorff space always a zero set?

I have been trying to prove that if $A$ is a closed set which is also an intersection of countably many open sets then $A$ is the zero set for some continuous real-valued function however have thus ...
3
votes
1answer
170 views

Can C([0,1]) be written as a countable union of compact sets?

Let $X=C([0,1])$ be the Banach space of continuous real valued functions on $[0,1]$ (with the $\sup$-norm). I am wondering if $X$ can be written as a countable union of compact sets $K_1 \subset ...
0
votes
1answer
35 views

Distance of functions to subspaces of $L^2$

Let $a>0$ and \begin{eqnarray} L_o&=&\{u \in L^2[-a,a]:\ u(-t) =-u(t)\ \text{ for a.e. } t\},\\ L_e&=&\{u \in L^2[-a,a]:\ u(-t) =u(t)\ \text{ for a.e. } t\}. \end{eqnarray} Find ...
5
votes
2answers
731 views

Unit ball in $C[0,1]$ not sequentially compact

This question is taken from Saxe K -Beginning Functional Analysis. Show that the closed unit ball in $C[0,1]$ is not compact by proving that it is not sequentially compact. (It's assumed that we ...
2
votes
1answer
69 views

Positive Continuous functions tending to $0$

Let $f(x) > 0$ be a member of $C(a, {\infty})$, that is, the space of continuous functions from the real number a to $+{\infty}$ . Suppose further that $f$ tends to $0$ as $x$ tends to $+{\infty}$. ...
2
votes
1answer
57 views

The Lebesgue Theory basic Application , get stuck

Ok, I am working on a very easy question but I get stuck when I trying to justify my answer. I know that, in order to use Lebesgue's dominated Convergence Theorem, there are two conditions that we ...
13
votes
1answer
650 views

Open Mapping Theorem: counterexample

The Open Mapping Theorem says that a linear continuous surjection between Banach spaces is an open mapping. I am writing some lecture notes on the Open Mapping Theorem. I guess it would be nice to ...
7
votes
2answers
166 views

Two-valued measure is a Dirac measure

Let $(X,\mathfrak B)$ be a measurable space such that $\{x\}\in \mathfrak B$ for all $x\in X$, and let $\mu$ be a positive measure on this space such that $$ \mu(B) \in\{0,1\} \quad\text{for all ...
6
votes
3answers
63 views

Exact sequence involving the nabla operator

Recently I noticed that $$0 \longrightarrow \Bbb R \overset{\text{const.}}\longrightarrow \mathcal{C}^\infty(\Bbb R^3,\Bbb R) \overset{\text{grad}}\longrightarrow \mathcal{C}^\infty(\Bbb R^3,\Bbb R^3) ...
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vote
1answer
65 views

$f \in \mathcal S, f(0)=1$ then $\lim_{\epsilon \to 0} f(\epsilon x) = 1$

Let $f \in \mathcal S(\mathbb R^n)$ with $f(0) = 1$. Here $\mathcal S$ means the Schwartz class. Then how can I prove that $$ \lim_{\epsilon \downarrow 0} f(\epsilon x) = 1 \; \text{(compact ...
1
vote
3answers
247 views

If $\Delta$ is the Laplace operator, what are $| \Delta |$ and $|\Delta +1|$

Assuming $\Delta : H^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ be the Laplace operator, then: What is the exact definition of $| \Delta |$? What is $|\Delta +1| $ also? This answer to another ...
8
votes
1answer
220 views

Dual space of space of all smooth function

On the space $C^\infty(S^1,\mathbb R)$, for each $n\in \mathbb N$, define $$p_N(\gamma)= \max\{|f^{(k)}(t): t\in S^1, k\leq N\}$$ Topology of all norms above define a metrizable locally convex ...
2
votes
1answer
284 views

Estimate on the norm of a self-adjoint operator

EDIT: thks to Martin's comment I realize the previous version was wrong. Here is the correct version of what I need to show: I am trying to show that if $A$ is a self - adjoint operator in a Hilbert ...
3
votes
2answers
203 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
2
votes
1answer
78 views

A reference request for sums of $C^*$-algebras

Does anyone know where I can find a reference for the following well-known fact: Let $(X_i)_{i\in I}$ be a family of compact Hausdorff spaces and let $X$ be the disjoint sum of all $X_i$s. Then ...
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vote
0answers
43 views

How can I prove $\mathcal S$ is dense in $W^{s,2}$?

Let $\mathcal S (\Bbb R^n)$ be the Schwartz class and $W^{s,2}(\Bbb R^n)$ be the Sobolev space($s=0,1,\cdots$). In fact I know that $C_c^\infty(\Bbb R^n)$ is dense in $W^{s,2} (\Bbb R^n)$ and ...
2
votes
0answers
71 views

Why is this true? (standard a-priori estimate on linear parabolic PDE)

I read this: We see that $$\frac{1}{2}\lVert u(T) \rVert^2_{L^2} \leq \frac{1}{2}\lVert u(0) \rVert^2_{L^2} + \lVert f \rVert_{L^2(0,T;H^{-1})}\lVert u \rVert_{L^2(0,T;H^1_0)}$$ and since $T$ ...
1
vote
1answer
78 views

Continuous Family of Norms&Operators

I want to ask the definition of a concept I see around but has not been able to find the definition for. Given a finite dimensional vector bundle $\pi : E \rightarrow K$ over a normed convex vector ...
0
votes
1answer
38 views

How to prove this equality in proportional fairness analysis?

How to prove: $$\sum_{s=1}^S\left(\frac{y_s-x_s}{x_s}\right)=\bigtriangledown J_\vec x\cdot(\vec y-\vec x)$$ with: $$J_\vec x=\sum_s\ln(x_s)$$
1
vote
1answer
128 views

proof of RKHS for a particular kernel is unique

Suppose that I have a kernel $K$. Then show that the RKHS $H_1$ and $H_2$ of $K$ are the same. So I need to prove the above statement. To begin with, as an exercise, I proved the reverse statement ...
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vote
1answer
132 views

Matrix Trace representation?

For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
2
votes
2answers
134 views

Example of Function $u\in H^{-1}(\Omega)\setminus L^2(\Omega)$

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $H^{-1}(\Omega)$ denote the dual of the Sobolev space $H_0^1(\Omega)$. Note that $$H_0^1\subset L^2\subset H^{-1}$$ How to construct a function ...
1
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1answer
46 views

there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each ...
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vote
0answers
189 views

Integral operators with operator valued kernels

This is the definition for integral operators I know: Let $\Omega \subset \mathbb{R}^n$ and $D \subset \mathbb{R}^n$. Let $K : \Omega \times D \to \mathbb{C}$ be measurable. A linear operator $T: ...
1
vote
3answers
311 views

Nonconstant linear functional on a topological vector space is an open mapping

In the middle of another proof (Theorem 3.4, p. 60) in his Functional Analysis book, Rudin says that "every nonconstant linear functional on $X$ (topological vector space) is an open mapping." Is ...
2
votes
2answers
124 views
0
votes
1answer
193 views

Essential Supremum

For $f\in L^\infty[a,b]$, show that $$\|f\|_\infty = \min \big\{M : m\{x \in [a, b] : |f(x)|>M\} = 0\big\}\;,$$ and if, furthermore, $f$ is continuous on $[a, b]$, that $\|f\|_\infty = ...
1
vote
1answer
124 views

Uniform Convergence of derivative

Question Suppose a sequence of derivatives of functions $f'_n $ converge uniformly to $f'$ where $f_n$ is defined on the on the interval $[a,b]$. And $f_n(x)$ converges pointwise to $f(x)$ for $x\in ...
4
votes
1answer
55 views

A question about operator representation

Let $H$ be a separable Hilbert space and let $A$ be a compact operator acting on $H$. In general we may write $H = E_A\oplus E_A^\perp$. Let us consider the $2\times 2$ operator matrix of $A$ ...
5
votes
1answer
501 views

On every infinite-dimensional Banach space there exists a discontinuous linear functional.

On every infinite-dimensional Banach space there exists a discontinuous linear functional. Assuming the axiom of choice, every vector space has a basis. With an infinite basis, I can define on a ...
2
votes
1answer
55 views

Why does the derivative have this form?

I will write the setup of the problem, but I don't think all the parts are necessary to answer my question. If you want the reference, this is from Rabinowitz : Minimax Methods in Critical Point ...
7
votes
1answer
344 views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
3
votes
1answer
152 views

$\ell^\infty$ and $\ell^1$

Show that $\ell^\infty$ and $\ell^1$ are normed linear spaces. Solution: Since $\ell^p$ is the collection of real sequences $a=(a_1,a_2, ... )$ for which $\sum_{k=1}^{\infty} |a_k|^p < ...
1
vote
3answers
273 views

Is it true that the unit ball is closed in a normed linear space iff the space is finite-dimensional?

I wonder about the following statement: Unit ball closed in a NLS iff the space is finite. Is this statement true? How would I go about proving this? I don't want to use that every norm in a ...
2
votes
0answers
184 views

Continuous, Bounded Normed Linear Spaces

For $f$ in $C[a, b]$, define $||f||_1 = \int_a^b |f|$. Show that this is a norm on $C[a, b]$. Also, show that there is no number $c\geq0$ for which $||f||_{\max}\leq c||f||_1$ for all $f$ in ...
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vote
1answer
1k views

Any two norms equivalent on a finite dimensional norm linear space.

I am trying to understand the proof that every two norms on a finite dimensional NLS are equivalent. I am working with this proof I found on the web: ...
6
votes
1answer
232 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
0
votes
2answers
73 views

An inequality $\| f \|_{L^p} \leq \| f \|_{L^\infty}^{1 - \frac{2}{p}} \| f \|_{L^2}^{\frac{2}{p}}$

What is the name of this inequality $$\| f \|_{L^p(\Bbb R^n)} \leq \| f \|_{L^\infty(\Bbb R^n)}^{1 - \frac{2}{p}} \| f \|_{L^2(\Bbb R^n)}^{\frac{2}{p}}$$ for $p > 2$?And how can I prove this?
0
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2answers
46 views

Hilbert Space - Question about norm

Let $H$ be a Hilbert space. Is it true that, if $\|x\|$ is less than or equal to $r$ and $\|y\|$ is strictly greater than $r$, then $\left\| x-\frac{ry}{\|y\|} \right\|$ is less than or equal to ...
2
votes
1answer
46 views

Basic Constrained functional form

Let $I \in C^1(\mathbb{R}^n,\mathbb{R})$ be an even functional. There is a claim that then the restriction to the sphere has the following form for the Frechet derivative $$I|_{S^{n-1}}'(u) = ...
9
votes
1answer
406 views

Distance minimizers in $L^1$ and $L^{\infty}$

If $H$ is a Hilbert space, we have the Hilbert Projection Theorem, which tells us that given a nonempty, closed, convex subset $K \subset H$, and a point $x \in H$, there is a unique point $y \in K$ ...
0
votes
1answer
197 views

Problem 5. ( chap3. p.87, functional analysis, W.Rudin)

I had done part a, b, and d,. But i cannot breakthrough part c, and part e,. I restate entired problem in the following: For $0<p<\infty$, let $l^p$ be the space of all functions $x$ (real or ...