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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7 views

can two Markov kernels be close in total variation and differ in their ergodicity properties?

This is a question inspired by a recent MCMC paper and linked to an earlier question of Sam Livingstone that did not get any answer. Given two Markov kernels $\mathfrak{K}$ and $\mathfrak{H}$ such ...
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1answer
15 views

Divergence of squared sum of Chebyshev Polynomials $\equiv L+R$ has empty point spectrum

The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation $$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$ Alternatively they are defined inductively: $$U_0(x)=1 ...
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2answers
29 views

The set of bounded continuous functions is a closed subspace of that of bounded functions

suppose S is a metric space and $B(S)$ is the set of bounded functions and $C_b(S)$ is the set consisting of bounded continuous functions. Prove that $C_b(S)$ is a closed subspace of $B(S)$. I ...
2
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1answer
55 views

Does $AB=(AB)^{\ast}$ and $A=A^{\ast}$ implies $B=B^{\ast}$?

Suppose that we have $AB=(AB)^{\ast}$ and $A=A^{\ast}$, does this implies that $B=B^{\ast}$? ($A^{\ast}$ is the Hermitian adjoint of $A$.) I have a feeling that they might not be equal in general. ...
3
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1answer
32 views

Prove a vector in $\ell^2(\mathbb{Z})$ is zero

Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$ That is, every elements of the vector is a series with the other terms in $\vec{c}$. ...
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20 views

Finding distance of $h(t)=t$ from a closed subspace $Y$ of $\pi$-periodic functions in $L^2(-\pi,\pi)$

Let $Y=\{f\in L^2(-\pi,\pi):f(t-\pi)=f(t) \text{for almost all $t\in(0,\pi)$} \}$ Show that there exists $g\in Y$ such that $$\|h-g\|_2=\inf \{\|h-f\|_2:f\in Y\}$$ where $h(t)=t$. Compute ...
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4answers
61 views

Does a definite integral define a linear functional?

Would $\displaystyle\int_0^1 t^2x(t)\,dt$ be a linear functional? For each $x$ in $P$ the function $y$ is defined by $\displaystyle\int_0^1 t^2x(t)\,dt$. I have to show that $y(ax+bz) = ay(x)+by(z)$. ...
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1answer
28 views

$L^2$ convergence by the sequence of domain

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Assume $u\in L^\infty(\Omega)$. We know a sequence $u_n\in L^\infty(\Omega)$ such that $$ \sup_{n}\|u_n\|_{L^\infty}<+\infty $$ ...
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2answers
38 views

Continuity of a function between metric spaces

I want to show: Let $(X,d)$ be a metric space and $A \subset X$ be a closed subset. Define $f: X \to \mathbb{R}$ by $$ f(x) = d(x,A) := \inf_{y\in A}d(x,y), \phantom{.} \forall x \in X.$$ Show ...
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37 views

Suppose $\{f_n(x)\}$ is a sequence in $C[a,b]$ such that for every $x \in [a,b]$ there exists a $M_x$ such that $|f_n(x)|<M_x, n\in \{1,2,3,\cdots\}$

Suppose $\{f_n(x)\}$ is a sequence in $C[a,b]$ such that for every $x \in [a,b]$ there exists a $M_x$ such that $|f_n(x)|<M_x, n\in \{1,2,3,\cdots\}$ Prove that there exist a $c,d \in [a,b]$ with ...
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1answer
20 views

Proof $\mathcal{C}^1(-1,1)$ is not a closed subspace of Sobolev space $H_0^1 \left[-1,1\right]$

Give a sequence of functions $\varphi_n\in \mathcal{C}^\infty(-1,1)$, Cauchy with respect to the Sobolev space $H^1_0$ norm $$|| \varphi||_1=\sqrt{\int_{-1}^1 (\varphi')^2+\int_{-1}^1 \varphi^2}$$ ...
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0answers
11 views

Can we attach a space with discrete signal?

This question refers to the link https://en.wikipedia.org/wiki/Space_(mathematics) and https://en.wikipedia.org/wiki/Discrete-time_signal. My question is how can we associate a discrete signal with a ...
3
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0answers
35 views

Convergence of average of translates of a function

Short version for people who don't like reading: Let $f\colon\mathbb{R}\to\mathbb{R}$ be $1$-periodic and $L^1$ on one period (or perhaps: measurable and bounded). Is it true that, for almost ...
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2answers
37 views

showing projection is a linear operator

Show that the orthogonal projection is linear. Let $x_i=y_i+z_i$, where $x_i\in X$, $y_i\in Y$, $z_i\in Y^\perp$, and $\alpha,\beta$ be scalars. Then \begin{align}P(\alpha x_1+\beta ...
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1answer
36 views

How to calculate second norm in functional analysis? [on hold]

I want to know second norm of $||x-y|| ^ 2$ is the same as $(x-y) ^ 2$?
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1answer
12 views

If $W = \{ (y_{n}) \in c_{0} : \ y_{1}+y_{2}+y_{3}=0\}$ what is $\operatorname{dim}(c_{0}/W)$?

Let $c_{0}$ be the space of sequences which converge to $0$. Let $W$ be a subspace of $c_{0}$ defined as $$W =\bigl\{ \{y_{n}\} \in c_{0} : \ y_{1}+y_{2}+y_{3}=0\bigr\}$$ How does one calculate the ...
3
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2answers
37 views

Intuition behind: Integral operator as generalization of matrix multiplication

So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications. In ...
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0answers
20 views

Epsilon delta proofs of theorems of continuity

Can anyone suggest a book which contains epsilon delta prooves for properties and theorems of continuity rather than sequential proofs.
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0answers
15 views

Self-adjoint extensions of symmetric linear differential operators

Are they general techniques to determine the sef-adjoint extensions of a symmetric linear differential operators in 2 variables with (real) variable coefficients? Any suggestions for references? ...
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18 views

distance in Hilbert space

W is closed and convex subset of Hilbert space $H$. Suppose $x\in H$, $x_0\in W$. I am trying to proof, that $d(x, W) = ||x-x_0||$ iff $\Re(<x-x_0, y-x_0>)\le 0$ for all $y\in W$. Could anyone ...
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0answers
21 views

How do we prove that a specific kernel is positive definite (case of logarithm)?

I have a problem proving that some specific kernels are positive definite. In general, I can find the answer quickly enough but here I have a specific case involving a logartihm : ...
2
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1answer
37 views

Not all norms are equivalent in an infinite-dimensional space

How to prove that not all norms are equivalent in an infinite-dimensional vector space? In particular, I would like to prove that for a space $X$ of continuous real-valued functions defined on ...
2
votes
1answer
30 views

A relation between two properties of sequences of operators

We have $(T_l)_l$ a sequence of bounded linear operators from $\ell^2$ to $\ell^2$. $\bullet$ We say $(T_l)_l$ satisfies the property "A" if ...
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0answers
15 views

Triangle inequailty for $L^p$ norm to power $p$

I would like to prove the sharp estimate for the $L_p$ norm to power $p$ with $1\leq p <\infty$. What is the constant $C$ here: $$\left\|\sum_{j=1}^Jf_j\right\|^p_p\leq C\sum_{j=1}^J\|f_j\|_p^p$$ ...
1
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0answers
21 views

Closed subset for $L^2$ strong and weak convergence

I was trying to solve the following exercise. Let $K$ a closed subset of $\mathbb{R}$. $$X:=\{f\in L^2[0,1]:f(x)\in K \:a.e.\:x\in [0,1] \}$$ Then: 1)X is closed under strong convergence in $L^2$. ...
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22 views

meaning of a dense subset

I'm trying to understand something - say I want to prove a certain property of functions in a space X. Is it enough to prove this property over functions which belong to a dense subset of X?
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1answer
68 views

What is the main purpose of learning about different spaces, like Hilbert, Banach, etc?

I just started to learn about functional analysis and have started to learn about various spaces, like $L^{p}$, Banach, and Hilbert spaces. However, right now my understanding is rather mechanical. ...
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0answers
16 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
3
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0answers
26 views

Riesz-Type Representation Theorems for Convex Functionals

It is well known that any positive linear functional $L$ on the spase $C_c([a,b])$ of functions continuous on an interval $[a,b]$ with compact support can be written as \begin{align*} ...
4
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3answers
54 views

A non continuous linear map $A:\Bbb{R}[X]\rightarrow \Bbb{R}$ such that $A(P)=P(1).$

I have a linear map $A:\Bbb{R}[X]\mapsto \Bbb{R}$ such that $A(P)=P(1)$, for the $p-$norm : $\Vert P\Vert=\bigl(\sum_{i=1}^n\vert a_i\vert^p\bigr)^{1/p}$ where $p\in[1,+\infty].$ For the cas ...
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0answers
29 views

show that $f_{\epsilon} \in D(\Omega)$; moreover, $f_{\epsilon} \to f$ uniformly as $\epsilon \to 0$.

Let $K$ be a compact subset of $\Omega \subset \mathbb{R^m}$, $\Omega$ is open and nonempty and let $f \in C(\Omega)$ have support contained in $K$. For $\epsilon \gt 0$, let ...
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3answers
42 views

proving that a function is well defined

Please, can someone help me? I have the following problem: Let $X$ be a normed space, $Y \subset X$ a linear subspace of $X$ and the function $$d_{Y}(x)=inf\{||x-y||:y \in Y\}$$;I have to prove that ...
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1answer
18 views

question about weierstrass approximation theorem true or false justify [on hold]

Is the following assertion true or false? There exists a nonzero function $f \in C([0,1])$ such that $$\int_0^1f(x)x^ndx=0 (\forall n \in \mathbb N)$$ holds. (Hint: use the weierstrass approximation ...
0
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0answers
29 views

Fourier transform calculus of tempered distributions

For example I wanted to ask confirmation of this calculation, if $u \in \mathcal{S}'(\mathbb{R}^n)$ then $\widehat{D^\alpha u} =(2\pi i \xi)^\alpha \widehat{u}$. By definition $\langle \varphi , u ...
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1answer
25 views

What is the difference between “exclusively depends” and “only depends”?

What is the difference when someone says that an expression exclusively depends on $x$ and an expression only depends $x$?
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1answer
15 views

If $f(r,t)=g(r),\,\forall r,t$ would that make $f$ and $g$ constant functions?

I know that if $f(r)=g(t),\,\forall r,t$ then $f(r)=g(t)=constant$, but If $f(r,t)=g(r),\,\forall r,t$, where now $f$ depends on $t$ would that lead to the same conclusion i.e. ...
1
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1answer
16 views

Application of uniform boundedness principle for sequence of operators

Let $T_n:X:=C([0,1])\rightarrow C([0,1])=:Y$ be a sequence of linear operators. Suppose: $T_n$ is bounded $(Tf)(x):=\lim\limits_{n\to\infty}(T_nf)(x)$ exist for every $f\in C([0,1])$ and $x\in ...
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1answer
27 views

A faithful positive Radon measure

Let $X$ be a locally comapct and Hausdorff space. We say a positive Radon Measure on $X$ is faithful if $$0\leq f ~~~,~~~\int fd\mu=0\rightarrow f(x)=0 ~~\forall x\in X$$ Q: True or false: If there ...
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0answers
20 views

Support of a Radon measure

Let $X$ be a locally compact and Hausdorff space. For a given Radon measure $\mu$ on $X$, the support of $\mu$ is the smallest closed subset of $X$ with $|\mu|(X)=\lVert\mu\rVert$ (where $|\mu|$ is ...
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1answer
15 views

Nonlinear operator sends bounded set to relatively compact set

Consider $g$ a continuous function on $[a,b]\times\mathbb{R}$, and let $z_0\in\mathbb{R}$. Define the (nonlinear) operator on $C[a,b]$: $$Mv(x)=z_0+\int_a^x g(t,v(t))\,dt$$ for $x\in[a,b]$. Prove ...
3
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1answer
22 views

Show that $f_{n}^2(x)$ does not converge in $D^1({\Omega})$

Let $$ f_n(x) = \left\{ \begin{array}{ll} n & \mbox{if $0 \lt x \lt \frac{1}{n}$};\\ 0 & \mbox{otherwise}.\end{array} \right. \ $$ I have to show that $\lim_{n \to ...
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1answer
14 views

Infinite dimensional $\sigma$-compact space

It is well-known that Banach space is $\sigma$-compact iff it is finite dimensional. I'm looking for examples of infinite dimensional normed $\sigma$-compact spaces.
1
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1answer
32 views

distance on a normed space

Please, can someone help me? I have the following problem: Let $X$ be a normed space, $Y \subset X$ a linear subspace of $X$. I have to prove that the function $$d_{Y}(x)=inf\{||x-y||:y \in Y\}$$ is a ...
1
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2answers
32 views

Lp space Inclusion Examples

I proved for a bounded set $\Omega$ and $1 \leq p \leq q \leq \infty$ that $L^{q}(\Omega) \subset L^{p}(\Omega)$. What is an example that would show strict inclusion, $ p<q$, and false if $\Omega$ ...
2
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1answer
26 views

A subspace of a mapping space?

We have a set $$ M=\{f:\mathbb{R} \rightarrow \mathbb{R}\mid f(1)>0\}\;.$$ I have never encountered this kind of set before. I assume it is correct to say that $M$ is a subspace of a mapping ...
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1answer
17 views

Why the dual to $c_0$ is $l^1$ and the space of sequences with bounded partial sums?

The dual to $c_0$ is $l^1$, but if $\{x_n\}_{n\in\mathbb{N}}\in c_0$, than according to Dirichlet's test $\sum_{n\in\mathbb{N}}(-1)^nx_n$ converges. But $\{(-1)^n\}_{n\in\mathbb{N}}\notin l^1$. So why ...
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1answer
44 views

Question about the weak solution of $u''-u=f$

I have a question about the ODE (weak formulation) given by $$u''-u=f$$ where $u\in H^1(\mathbb{R})$ and $f\in L^2(\mathbb{R})$. I want to see if there is an explicit formula for the weak solution. ...
0
votes
2answers
27 views

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n\}$ is closed

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n \}$ is closed. I had to show it is compact, and I am done showing it is relatively compact, but now I am stuck showing it is closed. ...
3
votes
1answer
19 views

States in a $C^*$-algebra bounded?

A functional $\phi$ on a $C^*$-algebra $A$ with unit element, i.e. $\phi: A \rightarrow \mathbb{C}$ is called a state if $\phi(T^*T) \ge 0$ for all $T \in A$ and $\phi( \operatorname{id}) = 1.$ Now, I ...
0
votes
0answers
23 views

What kind of information does the derivative of a sequence of functions give us?

When we derivate a function we know for example when then function increases and drecreases but when I derivate a sequence of functions I don't how to interpret the derivative since it is a collection ...