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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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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16 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
20 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
12 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
13 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
9 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
9 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
11 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
9 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
12 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 ...
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1answer
21 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 ...
1
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2answers
34 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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0answers
7 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 ...
1
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1answer
22 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
25 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
11 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
18 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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0answers
25 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
24 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
6 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
34 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
22 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
34 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
16 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 ...
0
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0answers
19 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 ...
0
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1answer
39 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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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
14 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 ...
1
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1answer
42 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
23 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
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1answer
43 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 ...
1
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1answer
12 views

Interior of a set in a normed space

Let $X$ be a normed space, $f \in X'$, $\alpha \in \mathbb{R}$; Prove that: $$\text{int}\{x \in X: f(x) \ge \alpha\}=\{x \in X:f(x)>\alpha \}$$ I noted the set in the left by $A$ and the set in the ...
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0answers
25 views

closed subspace of a linear space

Please, can somebody explain something to me? What exactly does it mean that a subspace $Y\subset X$, where $X$ is a normed space, is closed? It means that every convergent sequence of elements in $Y$ ...
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0answers
24 views

Matrix Inequality for the identity and a traceless matrix

Given a traceless matrix $C$ $\in M_n(\mathbb{F})$, i.e., tr$(C)=0$, what is the relationship between tr$|\mathbb{I}+C|$ and tr$|C|$? The two matrices are of dimension $n$. This was cross-posted to ...
1
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1answer
36 views

Sub-linear functional $f$ satisfies $f(0)=0$

Let $f$ be a sub-linear functional on a real vector space $X$ satisfying $f(x)+f(y)\ge f(x+y);f(\alpha x)=\alpha f(x);\alpha>0$ Show that it satisfies $f(0)=0 $ Thoughts: Now $f(x+y)\leq ...
4
votes
1answer
52 views

Check the proof of $||x||^2$ is not a norm

Show if $f$ is a norm: For $\mathbb{R}^n$, Define $f: \mathbb{R}^n \rightarrow \mathbb{R} $ by $ f(x) = \|x\|^2$ =$\sum_{n} x_n^2 $ I tried to solve if $f$ satisfies the three properties of a norm: ...
3
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1answer
26 views

Does a contractible set have contractible preimage, under a linear map?

Let $T:V\to W$ be a linear map of vector spaces, and let $A\subset W$ be contractible. Then is $T^{-1}(A)$ also contractible?
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56 views

About the gradient of a function in $H^{1}(\Omega)$

let $\Omega \in \mathbb{R}^{n}$ a bounded domain and $u \in H^{1}(\Omega)$ a real function. In the Leoni's Book - A First Course in Sobolev Spaces, the author define $\nabla u = (D_{1} u,\dots, ...
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2answers
25 views

About the definition of $L^{\infty}$ norm

Let $\Omega$ a limited domain in $\mathbb{R}^{n}$, the space $L^{\infty}(\Omega)=\{f: \Omega\to\mathbb{R} $ measurable $; ||f||_{L^{\infty}(\Omega)}<\infty\}$. Then if a function $f \in ...
2
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2answers
35 views

Prove norm inequality

It is given that $$\left\lVert x-y\right\rVert =\left\lVert y-z\right\rVert = \left\lVert z-x\right\rVert \qquad (1) $$ where $x,y,z \in \Bbb R^2$ and $ \left\lVert x\right\rVert=\sqrt ...
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0answers
22 views

Inequality in Banach space [duplicate]

So I have to either prove or disprove this inequality: $$ \left\lVert x\right\rVert^2 - \left\lVert y\right\rVert^2 \le \left\lVert x-y\right\rVert \left\lVert x+y\right\rVert$$ I know this to be ...
0
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1answer
14 views

problem on annihilators on finite dimensional spaces

Suppose $V$ and $W$ are subspaces of a finite-dimensional vector space $U$. Show that if $V^0 \subset W^0$ then $W \subset V$ This is an exercise problem in Linear Algebra Done Right, 3rd ...
3
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2answers
32 views

Composition involving bounded linear operators

I recently come across the following statement mentioned in a proof: Let $X,Y$ be normed linear spaces and $T:X \rightarrow Y$ be a linear operator. if for every bounded linear functional $U: Y ...
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39 views
+50

how to prove this function has zeros interlacing and including those of Riemann zeta

Let $$\chi (t) = \dfrac{4 i \pi \zeta (t) \left( \left( \ddot{\Psi} \left( \frac{t}{2} \right) - \ddot{\Psi} \left( \frac{1}{2} - \frac{t}{2} \right) \right) \zeta (t)^3 - 48 \zeta (t) \dot{\zeta} ...
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0answers
26 views

Convex function from condition

Can we deduce that $F$ is a convex function (i.e $F''(t)>0, \forall t>0$) from the following conditions: $F(t)=\int_{0}^t f(\xi) d\xi$, $0\leq \theta F(t)< t f(t), \forall t>0$ The ...
1
vote
3answers
42 views

application of the inequality $\|fg\|_1 \leq \|f\|_p\|g\|_q$

application of the inequality $\|fg\|_1 \leq \|f\|_p\|g\|_q$ where $1/p + 1/q = 1$ I know this is a straight application of the inequality, but how am I assured that the integral of ...
0
votes
1answer
11 views

infimum of operator norms of iterations of linear operators

I am currently reading a proof in which a fact is used without proof: For a Banach space $X$ and a bounded linear operator $T: X \to X$, $$ \lim_{n \to \infty} \| T^n \|^{\frac{1}{n}} = \inf_{n ...
0
votes
1answer
26 views

When a set is convex, how does the polar set of its polar set equal the original?

I have read the following proposition, and haven't been able to connect the convexity of $X$ to the statement's main equality. Any guidance would be much appreciated. Define $X^\text{o}$, the polar ...
-1
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0answers
18 views

Prove dimension finiteness for a separable subspace of $L^\infty(0,1)$. [on hold]

Let $X$ be a separable subspace of $L^\infty(0,1)$. How do I prove that $X$ is finite dimensional? Thanks in advance.