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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14
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1answer
298 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
0
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0answers
46 views

Continuous non-differentable functions

I'm looking for some examples of everywhere continuous functions which are nowhere differentable. I found already Takagi curve and Weierstrass function. Can you point out some online courses or pdf ...
2
votes
2answers
50 views

Prove that there is no norm for to make this mapping continuous

I am dealing with an exercise which is as follows: Show that there is no norm such that the set of all the mappings $T_a$ which map every element $f\in C(\mathbb{R}, \mathbb{R})$ (where the latter is ...
1
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2answers
30 views

Orthonormal projection contracts inner product?

I wonder if an orthonormal projection $P^2=P$ in a Hilbert space $\mathcal{H}$, contracts its inner product i.e. $\langle PW,V \rangle \leq \langle W,V \rangle $ for every pair of elements $W,V \in ...
2
votes
1answer
22 views

Image of a dense set through unbounded operator

Let $T$ be a densely defined, closed operator on a Hilbert space $H$ such that $T^*T$ remains densely defined. Obviously, $\sigma(I+T^*T)\subset [1,\infty)$, which in particular implies this operator ...
1
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0answers
33 views

Finding the general function that satisfies this property

I've been playing around with some introductory examples in my functional analysis course, and I came across. Given $X = \mathbb{R} \setminus \{0,1,2\},$ I want to find all functions $f: X ...
2
votes
1answer
37 views

$C ([1,2] \times [0,1] \to \mathbb R)$ dense in $C ( [1,2] \rightarrow L^{2} ([0,1] \to \mathbb R))$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
2
votes
1answer
38 views

Continuous functions with values in separable Banach space dense in $L^{2}$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
0
votes
2answers
35 views

Spectrum of operator $T((x_n)_{n\in\mathbb{Z}})=\left(\frac{1}{n^2+1}(x_n-x_{-n})\right)_{n\in\mathbb{Z}}$

The eigenvalues should satisfy: $$T(x_n)=\lambda x_n$$ $$\frac{1}{n^2+1}(x_n-x_{-n})=\lambda x_n$$ $$\left[(n^2+1)\lambda+1\right]x_n=x_{-n}$$ I suppose that this should mean that ...
2
votes
1answer
57 views

Pythagorean theorem does require the Cauchy-Schwarz inequality?

In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved ...
0
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0answers
145 views

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$.

Show that there is no normed space X s.t. its dual X' is equivalent to $((C^1[0,1];\mathbb{R}), ||.||_{\infty})$. Normed spaces are equivalent if there are bounded linear maps which are inverses of ...
2
votes
1answer
35 views

Show that if we had a complete metric space $X$ with no isolated points, then every singleton $\{x\}$ is nowhere dense

My attempt: The closure of the singleton is again the singleton itself Since there are no isolated points, then clearly $\{ x \}$ does not contain any non-empty open set hence the interior of the ...
0
votes
1answer
43 views

If we had a complete metric space with no isolated points, then singular points are nowhere dense

Let $X$ be a complete metric space. I am trying ot prove whether or not each point in $x$ is nowhere dense if $X$ has no isolated points. idea: the closure of a point is itself, and the interior of a ...
2
votes
1answer
42 views

Square Root of the shift operator indexed by $\mathbb{Z}$

My question is very similar to this question, but instead of indexing by $\mathbb N$ I am indexing by $\mathbb Z$. Consider the shift operator $T : \ell^1(\mathbb Z) \to \ell^1(\mathbb Z) $ given by ...
0
votes
1answer
35 views

How do we get the inequality?

Proposition: If $A \in \mathbb{R}^{n \times n}$ a symmetric matrix then $||A||= \sup \{ ||Ax||_2: ||x||_2=1\}= \sup \{ |\langle x, Ax \rangle|: ||x||_2=1\}$. Proof: It suffices to show that $||A|| ...
0
votes
0answers
17 views

For a convex function, why does lower semicontinuity imply weak lower semicontinuity?

I have come across the statement that, for a convex function, all notions of lower semicontinuity are equivalent. That is: weak lower, sequential lower, and weak sequential lower semicontinuity are ...
0
votes
0answers
23 views

Local banach algebra without zero divisors

I need to construct example of banach algebra with unique nontrivial maximal ideal and without zero divisors. I think it is must be a subalgebra of $\mathbb{C}[[z]]$, but I could not build anything.
2
votes
0answers
35 views

Conditions under which an Convolution operator is normal.

I have a possibly complex valued convolution operator given by $ \int_{\mathbb{R}}K(x-y)f(y)dy$ I know that the operator is self-adjoint if $K(x)=\overline{K(-x)}$ holds. But are there softer ...
1
vote
1answer
33 views

problem on finding norm

let $f$: $l^2$ $\to$ $\Bbb R$ be defined by $$ f(x_1,x_2,x_3,......) = \sum_{n=1}^\infty \frac{x_n}{2^\frac{n}{2}} \ \forall x=(x_1,x_2,....) \in l^2$$ then, what is the value of $\left\lVert ...
0
votes
1answer
26 views

Properties about reflexive space

I'm studying fuctional analysis and specifically reflexive spaces. My textbook has a introductory level, so don't cover so many things. My questions are: 1) If $X$ and $Y$ are isomorphics and $X$ is ...
2
votes
1answer
51 views

Properties of mollification

We have this theorem For any $1\le p<\infty$ and $f\in L^p(\mathbb{R}^k)$, then $\|f*\phi_\delta - f\|_p\to 0$ as $\delta\to0$, where $\phi$ is any nonnegative measurable function on ...
0
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0answers
26 views

On the sequence of orthonormal basic

I have a question : Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such ...
0
votes
1answer
18 views

How can we find a contradiction?

Let $(X, \rho)$ be a metric space and $x \in X, A \subset X (A \neq \varnothing)$. We have $x \in \overline{A}$ iff $d(x,A)=0$. We suppose that $d(x,A)=0$ . We want to show that $x \in ...
0
votes
1answer
20 views

Non spatial isomorphisms

Let $H$ be a Hilbert space. Any unitary operator $u\in B(H)$ induces an spatial isomorphism $\phi_u(x)=uxu^*$ on $B(H)$. Question: Let $\phi:B(H)\to B(H)$ be a surjective *-ismorphism. Is $\phi$ ...
5
votes
0answers
46 views

Minimum infinity norm control problem

I am having trouble understanding Example 2 of section 5.9 of Luenberger's Optimization by Vector Space Methods. The problem is to select a current $u(t)$ on $[0,1]$ to drive a motor governed by $$ ...
2
votes
1answer
17 views

Express the solution of the integral equation in the resolvent form

Express the solution of the integral equation $$f(x) = \phi(x)+\lambda\int_0^{2\pi} \cos(x+t)f(t) \, dt$$ in the resolvent form $$f(x) = \phi(x)+\lambda\int_0^{2\pi} \Gamma(x,t;\lambda)\phi(t) \, ...
2
votes
1answer
28 views

I do not understand a point in the proof of completness of $L^{\infty}$

do not understand a point in the proof of completness of $L^{\infty}$. I have this proof. We consider the sets $$A_{n,m}=\{x\in E:|f_{n}(x)-f_{m}(x)\|\leq\|f_{n}-f_{m}\|_\infty\}$$ for all ...
1
vote
0answers
13 views

Find the codimension of $\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $\ell_2$.

Find the codimension of $A=\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $l_2$ where $S$ is the shifting operator to the right: $Se_i=e_{i+1}$. I don't quote understand ...
1
vote
1answer
25 views

Examples of bounded linear operators with range not closed

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range? ...
0
votes
0answers
21 views

properties of vector space

Let $E$ be Banach space and $0<r<1$, $1\leq p<\infty$. Define the set $A$ as follow $$A=\left\{(x_j)_{j\in\mathbb{N}}\subset E :\sum\limits_{j=1}^{\infty}\left[\left|\left\langle ...
0
votes
1answer
24 views

Weak operator topology convergence of hermitian operators

Let $\{A_i\}$ be a net of hermitian operators on a separable Hibert space $\mathbb{H}$ and suppose that there is a hermitian operator T such that $A_{i}\le T$ for all i. If $\{<A_i h,h>\}$ is an ...
-1
votes
0answers
15 views

Is the distance attained?

Suppose that we consider the set $K:=\{ x \in \mathbb{R}^n: \sum_{j=1}^n |x_j|^p \leq 1 \}$ where $0<p<1$. In this case the set isn't convex. Indeed, if we pick for example $x=(1,0,0, \dots), ...
1
vote
2answers
35 views

Prove equality of norms of operators

Let $e_i$=${\{\delta_{k,i}}\}_{k\ge1}$ $\in$ $l_2$, $i\ge1$, $A_n$ and $B_n$ - operators that are defined like this: $A_n\{x_i\}_{i\ge1}$ = $x_ne_1$, $B_n\{x_i\}_{i\ge1}$ = $x_1e_n$ ...
0
votes
0answers
37 views

Reference request: about inverse Laplacian operator

I am currently studying some problems about inverse Laplacian and the Yosida approximation and wishing to learn more about it. Here is a post about one of the problems that I am interested in. ...
2
votes
1answer
21 views

property of the Gelfand transform: why does an isometry map closed sets to closed sets?

The following is a theorem about self-adjoint subalgebra of $C(X)$ where $X$ is compact Hausdorff and the first half of its proof: Here are my questions: Why $\Gamma:\mathfrak{U}\to ...
3
votes
1answer
62 views

Closure of an Operator in $l^2$

Let $l^2$ denote the Hilbert space of all complex sequences $\phi = (\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j|^2 < \infty$. Consider the linear subspace of $l^2$ defined by ...
0
votes
1answer
24 views

Determining isomorphism between sequence spaces

What's a good way to dis/prove that the $\mathbb{R}$-vector space of real convergent sequences and that of all real sequences are (linearly) isomorphic? The former space is isomorphic to $c_0$. Maybe ...
0
votes
1answer
20 views

continuity of a function and net convergence

The following is a statement and its proof in the Banach Algebra Techniques for Operator Theory by Douglas: I don't understand the last part of the proof. In order to show that $f$ is continuous, ...
0
votes
1answer
37 views

Show that for any function $f_0 \in L^p(E)$ there is a function $g_0$ in $C \subset L^p(E)$ closed convex

Let $E$ be a measurable set $1< p<\infty$ and $C$ a closed bounded convex subset of $L^p(E)$. Show that for any function $f_0 \in L^p(E)$ there is a function $g_0$ in $C$ for which ...
0
votes
0answers
21 views

Does this transformation(harmonic analysis) exist?

Assuming that there are two disjoint sets(A and B) of high dimensional $N^d$ integers. Each point $V$ can be expressed by a periodic function: $f(t)=v_1*cos(t) + v_2*sin(2t) + ...
0
votes
0answers
34 views

The inverse of Laplacian for different orders.

This question is related to my previous question here Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Let $\Delta$ denote the Laplacian operator, ...
0
votes
1answer
21 views

Form of elements in closed linear span

Let $H$ be a Hilbert space, and $\{x_j\}$ an orthonormal set in $H$. Let $C$ be the closed linear span of $\{x_j\}$. I am trying to prove the following: Let $x$ be an element in $C$. Then $x=\sum ...
0
votes
0answers
15 views

Normal positive functional on Von Neumann algebras

Let $A$ be Von Neumann algebra. A positive linear functional ‎$‎‎‎\varphi‎$ on $A$ ‎is ‎said ‎to ‎be ‎normal ‎if ‎for ‎any ‎self‎adjoint and increasing nets such that ‎$‎‎u_{\alpha‎}\longrightarrow ...
2
votes
1answer
34 views

Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
0
votes
2answers
113 views

Continuity of a function on a normed space

If $\mathbb{X}$ and $\mathbb{Y}$ are normed spaces, and E is a subset of $\mathbb{X}$ such that $f : E \rightarrow \mathbb{Y}$. How can I show that f is continuous if and only if for every closed ...
0
votes
2answers
50 views

Adjoint of an Operator in $l^2$

Let $l^2$ be the Hilbert space of all complex sequences $\phi =(\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j |^2 < \infty$. Set $D= \{ \phi \in l^2 : \sum_{j=0}^{\infty} j ...
0
votes
1answer
24 views

What is a generalized stochastic process? I've found two different definitions. Are they equivalent?

Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$. What is a generalized stochastic process? I've found two different definitions in some textbooks: ...
1
vote
1answer
21 views

How to check compactness of linear operator

Let the continuous linear operator $T:l^2\to l^2$ be defined by $$T(x_1,x_2...)=(0,x_1,0,x_3,0,x_5,0...).$$ Are $T$ and $T^2$ compact? A linear operator $T: X\to Y$ is said to be compact if image ...
0
votes
0answers
17 views

$C^{\infty}_{C}(\Omega)$ it is dense in $ W^{1,2}(\Omega)=H^1(\Omega) $ ? I know that it is dense in $H^1_{0}(\Omega).$

$ C^{\infty}_{C}(\Omega) $ it is dense in$ W^{1,2}(\Omega) = H^1(\Omega)$ ? I know that it is dense in $H^1_{0}(\Omega) $ .
0
votes
0answers
14 views

Homotopy of bounded homomorphisms between Banach algebras

Let $A$ and $B$ be Banach algebras. Say that two bounded homomorphisms $\phi_0$ and $\phi_1$ from $A$ to $B$ are homotopic if there is a path $(\phi_t)_{t\in[0,1]}$ of bounded homomorphisms from $A$ ...