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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The orthogonal projection of $\gamma(x)=2e^{2\pi xi}$ over the subspace generated by …

The orthogonal projection of an element $x_0 \epsilon$ H over a convex set C is the element $y_0 \epsilon$ C such that $\|x_0-y_0\|=\min_{y \in C}\|x_0-y\|$. Find the orthogonal projection of ...
0
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3answers
50 views

$∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$!

Hi I was thinking about a problem and have a question: we know that if $f∈C([0,1])$ for which $∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$! Now my question is: Do we still have the same when we ...
0
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1answer
22 views

there is a measurable function $f$ on $X$ such that $|{f(x)}|=1$ for a.a $x \in X$ and $\nu(E)=\int_Efd|{\nu}|$ for any $E \in \mathfrak{M}$

any hints on this problem: Let $\nu$ be a finite signed measure on a measure space $(X, \mathfrak{M})$ and let $|{\nu}|$ be its total variation, prove that there is a measurable function $f$ on $X$ ...
0
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1answer
38 views

Let $\{f_n\}_{n=1}^\infty$ be non-negative functions and $f_n \to f$ then $f \geq 0$

I have trouble with this question: Let $\{f_n\}_{n=1}^\infty$ be a sequence of non-negative functions in $L^2(0, 1)$, and suppose that $f_n$ converges to a function $f$ in the norm of $L^2(0, 1)$. ...
1
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1answer
13 views

Show that M is closed convex and find the minimum norm

Let M={$y=(y_1,...,y_n) \subset C^n: y_1+...+y_n=1$}. Show that M is closed, convex, and find the element of minimum norm in M. Prove M is convex Proof: A set $ M \subset C^n$ is convex if for every ...
2
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1answer
1k views

Volterra Operator is compact but has no eigenvalue

Volterra operator is defined as operator $V:L^2[0,1]\rightarrow L^2[0,1]$ by \begin{eqnarray} (V)(f(x))=\int_0^xf(y)dy \end{eqnarray} Would you help me to prove that this operator is compact but has ...
2
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1answer
30 views

sub-algebra of continuous real valued functions without unit must vanish at a point

If $X$ is compact Hausdorff and $A$ a closed subalgebra ( a vector space and closed under multiplication) of $C( X \to \mathbb R)$ the set of continuous real valued functions which separates points, ...
0
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0answers
20 views

Continuous compactly supported real valued functions on a locally compact and $\sigma$ compact space is separable

I already know that if $X$ is a compact metric space then the space of continuous real valued functions $C(X \to \mathbb R)$ are separable. What I'm trying to prove that if $X$ is a locally compact ...
0
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0answers
21 views

the sum of closed convex sets

Let C and D be two closed convex subsets of a Banach space with C+D is closed. If bounded sequence $\{x_n\}\subset C+D$, can we choose bounded sequences $\{c_n\}\subset C$ and $\{d_n\}\subset D$ such ...
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1answer
33 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
3
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1answer
18 views

Show that $\lambda \in \sigma(A),$ $\lambda$ not an eigenvalue, implies that $\lambda \in \sigma(A + K)$ where $K$ is compact.

Let $A : H \rightarrow H$ be a bounded linear map where $H$ is a Hilbert space with $\dim H = \infty$. Suppose that $\lambda \in \sigma(A)$ but $\lambda$ is not an eigenvalue. Let $K : H \rightarrow ...
2
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0answers
16 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
1
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1answer
23 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
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1answer
34 views

Exercise 3.6: Elementary Functional Analysis By Barbara [on hold]

Let $X=\ell_\Bbb R^\infty$ denote the space of bounded sequences with real entries, in the supremumnorm. Consider the operator $T$ defined on $X$ by $T(x_1, x_2, . . .) = (x_2, x_3, . . .)$; this is ...
1
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1answer
36 views

Fredholm operators in Hilbert spaces

Suppose $T_r$ and $T_l$ are the left and the right translations in $l_2$. $T_l$ maps $(x_1,x_2,x_3,...)$ to $(x_2,x_3,x_4,...)$, $T_r$ maps $(x_1,x_2,x_3,...)$ to $(0,x_1,x_2,...)$. It can be easily ...
2
votes
0answers
26 views

Subsequences of a basic sequence

Suppose ($x_n$) is a basic sequence in a Banach space $X$, and $Y$ is a closed, infinite co-dimensional subspace of $X$. Can we always find a subsequence ($y_n$) of ($x_n$) such that the intersection ...
1
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0answers
20 views

Strongly convergence in L^2

Bonjour, let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly converge to $u$ dans $L^2$ ...
2
votes
1answer
69 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
2
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1answer
35 views

The set $S=\{(x,y) \in \mathbb{R}^{n} \times \mathbb{R}^n = \mathbb{R}^{2n} ; x \neq y\}$ is connected if $n \geq 2$.

When n = 1 it is easy to see that is not connected, it just take the split open $ S=U_1 \cup U_2$ such that $U_1 = \{(x,y) \in \mathbb{R}^2 ; x > y\}$ is $U_2 = \{(x,y) \in \mathbb{R}^2 ; x < ...
2
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1answer
49 views

Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
0
votes
1answer
13 views

The orthogonal complement of $M \subset L^2(0,1)$ is the subspace generated by…

The orthogonal complement of $M \subset L^2(0,1)$ is the subspace generated by 1, $e^{2\pi ix}$, and $e^{4\pi ix}$. By definition the orthogonal complement of a subspace $M \subset H$ is the set ${y ...
0
votes
1answer
21 views

Isometry: Adjoint = Leftinverse

Given an isometric operator is it true that its adjoint is necessarily leftinverse? My attempt goes like this: $$\langle x,\mathbb{1}\tilde{x}\rangle=\langle x,\tilde{x}\rangle=\langle ...
1
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1answer
30 views

Does the open mapping theorem have a local version?

Let $T:X\to Y$ be a linear continuous surjection between Banach spaces $X$ and $Y$. By the open mapping theorem, we have $T$ is open. Now let $C$ be a closed convex subset of $X$ satisfying that ...
0
votes
1answer
39 views

Signed finite Radon measures with vague topology

If $X$ is a locally compact and $\sigma$-compact metric space. Let $M(X)$ be the space of signed finite Radon measures on $X$. (1) Show that measures with finite support is dense is $M(X)$ in the ...
11
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1answer
309 views

Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
0
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0answers
21 views

Proof about compact operator

I was in the process of proving If $X,Y$ are Banach spaces and $u \in B(X,Y)$ is a bounded linear operator then the following are equivalent: (1) $u$ is compact (2) for $S \subseteq X$ bounded, ...
1
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1answer
44 views

Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric

I've been struggling with this problem for the last four hours. The problem is to show that the space of $\mathbb{C}$-valued continuous functions on $[0,1]$ under the metric ...
-1
votes
1answer
49 views

Complexification the real inner product space

Let $V$ be a real inner product space. If $W=V\times V$ with the operations $(u_1,v_1)+(u_2,v_2)=(u_1+u_2,v_1+v_2)$ and $(\alpha +i\beta)(u,v)=(\alpha u-\beta v,\alpha v+\beta u)$, where $u, ...
1
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0answers
47 views

Proof of Arzela-Ascoli theorem

I tried to prove the Arzela-Ascoli theorem: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot ...
1
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0answers
15 views

How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
1
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0answers
17 views

Limit of exp of self-adjoint operator

Let $A$ be self-adjoint (possibly unbounded) operator on Hilbert space $\mathcal{H}$. Under what conditions $w-\lim_{t\rightarrow\infty} e^{i A t}=P_0$, where $w-\lim$ - the limit in weak operator ...
1
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0answers
27 views
+50

Gelfand transform on disk algebra

I tried to prove that if $A$ is the disk algebra then the Gelfand transform is the identity map. The statement can be found here in Theorem 4.4 but it is given without proof. Please can someone check ...
2
votes
1answer
55 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 ...
3
votes
1answer
29 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
6
votes
1answer
290 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" ...
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votes
1answer
29 views

Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...
4
votes
1answer
38 views

Is C(E)a dual of any linear norm space?

E is a closed bounded set of R. Is C(E)a dual of any linear norm space?
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0answers
18 views

Laplace transform of bochner integral

$\mathcal{H}$:real hilbert space with inner product $(\cdot,\cdot)$ and norm $||\cdot||:=(\cdot,\cdot)^{1/2}$ A family $(T_{t})_{t>0}$ of linear operators on $\mathcal{H}$ with ...
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0answers
28 views

Show that the map $T(x)=x/2+x^{-1}$ is a contraction and find $\alpha$

Let $X=[1,\infty)$. Show that the map $T(x)=x/2+x^{-1}$ is a contraction, and find $\alpha$. Proof: A function $T:X \rightarrow X$ is said to be a contraction if $dist(T(x),T(y)) \leq \alpha ...
0
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0answers
34 views

Is the martingale propertey preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
0
votes
1answer
35 views

Separability of functions with compact support

Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable? My ...
2
votes
1answer
18 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
0
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1answer
34 views

Question on sequence space (as a linear space)

Let $X$ be the space $\ell_\infty$ of all bounded sequences of real scalars. If $Y$ is the set of all $x\in X$ that have bounded partial sums (1) Can I say $Y$ is a linear space (as a subspace of ...
0
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1answer
23 views

Property of Projection Operator

Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex. I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
1
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1answer
22 views

Banach spaces: Convergence in terms of the Schauder basis.

Let $X$ be a Banach space. Suppose $X$ has a normalized Schauder basis $\{x_n\}_{n \in \Bbb N}$. Let $\{y_n\}_{n \in \Bbb N}$ be a sequence in $X$ converging to $0_X$. For each $n \in \Bbb N$, let ...
0
votes
1answer
15 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
1
vote
0answers
35 views

Open Convex Subsets of Dense Spaces

So I asked this question yesterday, Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric, and it made my start wondering the following... Suppose the following: ...
0
votes
0answers
19 views

use Fatou theorem to prouve an convergence

let $u_n$ an sequence uniformaly bounded in $H^1_0(\Omega)$, then, $u_n$ converge weakly to $u$ in $H^1_0$, and strongly in $L^2(\Omega)$ and a.e $x \in \Omega$. Let $g(x,u)$ an Carathedory function ...
63
votes
18answers
6k views

Your favourite application of the Baire Category Theorem

I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications ...
0
votes
0answers
49 views

Real Analysis and dynamics

I am looking for a textbook or similar resource that addresses the content in a rigorous graduate course in real analysis(at the level of Rudin/Royden) with the following criteria: No hand waving - ...