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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8 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
13 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*} ...
3
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2answers
37 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
22 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
39 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
16 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 ...
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0answers
20 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
23 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
14 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. ...
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1answer
15 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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13 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
17 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
12 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
20 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
12 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.
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1answer
30 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
23 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
21 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
15 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
39 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. ...
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2answers
24 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
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1answer
14 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 ...
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0answers
20 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 ...
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0answers
13 views

Dense convex set in $*$-weak topology

Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in ...
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0answers
13 views

Preservation of completeness through a continous onto mapping

Let $(X_{1},d_{1})$ and $(X_{2},d_{2})$ be metric spaces and $f: X_{1} \to X_{2}$ be a continuous onto map such that $$ d_{1}(x,y) \leq d_{2}(f(x),f(y)) \hspace{2mm} \forall\phantom{i}x,y \in ...
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1answer
15 views

Proving that a set infers a norm given certain conditions

Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff $\mathbb{N} . \bar{B} = E$ $\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in ...
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1answer
11 views

supremum of a function in a normed space

Let $X$ be a normed space and $A \subset X$. Prove that: $$\sup(f(A))=\sup(f(cl(A))=\sup(f(conv(A)),$$ where $f \in X^{*}$. For the first equality, I thought to prove it by double inequality: Let ...
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0answers
21 views

Distance from image of bounded operator

Let $A:H\rightarrow H$ be a bounded linear operator on Hilbert space $H$. Suppose we have $x\in H$ and $r>\mathrm{dist}(x,A(H))=\inf\{\|x-Ah\|,\ h\in H\}$. How to prove that then there exist ...
0
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1answer
25 views

Bound the variation of this decreasing function

Let $f(x)$ be a decreasing function defined over the interval $[0,a]$, with $f(0)=b$. The first derivative of $f(x)$, which is negative, is such that $f^\prime(x) > g(x)$, or equivalently ...
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2answers
37 views

Norm of an operator on space of real polynomials

Let $L:\mathbb{R}[X]\rightarrow\mathbb{R}[X]$ be an operator given by the following formula $L\left(\sum\limits_n a_nX^n\right)=\sum\limits_n a_{2n}X^{2n}$. We assume that on $\mathbb{R}[X]$ we have ...
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1answer
12 views

Bounded neighbourhood of zero in TVS

Is is true that in any topological vector space, which is $T_1$ there exists bounded neighbourhood of zero ? Is is still true if we omit $T_1$ axiom ?
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8 views

Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
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1answer
24 views

Weak convergence in $L^p$ equivalent to pointwise almost everywhere convergence

Can weak convergence of a sequence $f_n\in L^p(\Omega, \mu)$ to some $f\in L^p(\Omega, \mu)$ be characterised as almost everywhere pointwise convergence? Let us also assume the measure space is ...
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2answers
26 views

Weak convergence in Banach space

If $X$ is a Banach space, $X'$ its dual and $x,x_n\in X$ and $x',x_n'\in X'$, then the following implication holds: $x_n\to_w x$ (weak convergence) and $x_n'\to x'$ in $X'$ $\implies$ ...
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0answers
16 views

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well.

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well. Let $f,g \in C(\Omega)$. Since $f\ne g$, there is a $x_0 \in \Omega$ such that $f(x_0)\ne g(x_0)$. Hence ...
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1answer
19 views

a function defined on $c_{0}$

Can somebody help me ,please? I have the following function: $$\phi:c_{o}\to \mathbb{R}$$, $\phi(x)=\sum_{n\ge 1}(-2)^{1-n}x_{n}$, where $c_{0}$ is the set of the sequences of real numbers convergent ...
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1answer
34 views

Show a function is $K$-Lipschitz

The following is the proof taken from Lemma $13$. Questions: $(1)$: What is the Lipschitz-norm of $\phi_a$? The following is my attempt: $\| \phi_a \|_{Lip} = \sup_{x \neq y}{\frac{|f(a) - ...
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0answers
26 views

Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$

Let $f$ be a generalized function on $\Omega$. Let $G$ be an open subset of $\Omega$. Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$ Let $\phi ...
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0answers
11 views

What it means for a generalized function to be periodic or radially symmetric??

Let $T$ be a generalized function. I need to provide definitions for $T$ to be periodic and radially symmetric. A function (on $\mathbb{R})$is said to be periodic if there exists a $p \in ...
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0answers
41 views

Properties of weakly convergent series in Hilbert space

Let $H$ be a Hilbert space and $\{x_n\}_{n=1}^{\infty}$ given sequence of vectors from $H$. Suppose that for every $\{\alpha\}_{n=1}^{\infty}\in \ell^2$ series $\sum_{n=0}^{\infty}\alpha_nx_n$ is ...
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0answers
25 views

Linear Algebra Vector Space and Subspace [on hold]

If $X$ be an infinite dimensional vector space and $Y$ is subspace of $X$, then show that whether dimension of $Y$ is always finite or infinite also. Also give example of any subspace whose dimension ...
2
votes
1answer
37 views

Completeness of bounded linear maps

Let $X,Y$ be normed vector spaces over $\mathbb{C}$, and $L(X,Y)$ the space of all bounded linear maps from $X$ to $Y$. Its known that $L(X,Y)$ is a normed(operator norm) vector space. Theorem: ...
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1answer
19 views

Proof a real functional is continuous in $C_{[a,b]}$ (verification)

I wish to have some feedback on the following proof of the claim below, either if it is correct, what to fix, or other suggestions. Claim: Let $\psi :[0,1] \times \mathbb{R} \to \mathbb{R}$ be a ...
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1answer
26 views

Weak convergence of measures and compact sets

Suppose that we have a sequence of probability measures $\{ \mathbb{P}_n \}$ converging weakly to a probability measure $\mathbb{P}$. Suppose that $M$ is a metric space with a compact subset $K$. I ...
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1answer
41 views

Does a bounded sequence in $C^k$ have a convergent subsequence in $C^{k-1}$

Let $K \subset \mathbb{R}^n$ be compact, let $k \in \mathbb{N}$. Let $\{f_n\} \subset C^k(K)$ be a bounded sequence w.r.t $C^k$ norm. Does it have a convergent subsequence in $C^{k-1}(K)$. If so how ...
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0answers
15 views

Space of polynomials as a continuous image of F-space

Let $X=\mathbb{R}[a,b]$. Is there any norm $\|\cdot\|$ on $X$ s.th. $X$ is a continuous image of some $F$-space. ($F$-space means that there exists complete metric s.th. $d(x+z,y+z)=d(x,y)$) ? My ...
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0answers
15 views

Weak derivative of a piecewise defined function

I am currently looking at these online notes on PDEs, page 59. How does it follow that if $f^R = \phi(x/R) f(x)$ $ \phi(x) = \left\{\def\arraystretch{1.2}% \begin{array}{@{}c@{\quad}l@{}} 1 ...
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1answer
43 views

If $f$ is not continuous then $\ker f$ is dense in $X$

Let $X$ be a normed space and $f:X\rightarrow \mathbb R$ a linear function. I saw an old post with this problem, but there is not a complete proof. For beginning I have to consider that ...
0
votes
1answer
24 views

Differentiating a function composition

Given $g:R^n \rightarrow R^k$ and $h:R^k \rightarrow R$, we have $f(x) = h(g(x))$. Using the chain rule, we can differentiate $f(x)$ to get $f'(x) = \nabla^Th(g(x))g'(x)$ My question is why do we ...
2
votes
1answer
45 views

Proving that a function grows faster than another

I'm told to prove or disprove that $4^{\sqrt{n}}$ grows faster than $\sqrt{4^n}$ As n tends to infinity. From my Previous years Calculus I know that if I take the derivative of two functions, and one ...