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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1answer
26 views

What does the graph of a norm look like

I talking about functions of the form ||X||^p, when p>1 for different values of p. I know these are all convex functions, but I don't know how to graph them.
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4answers
52 views

Is the sum of an infinite series of elements in the span of an orthonormal set also in that set?

If $(e_k)$ is an orthonormal sequence in some Hilbert space $H$ does it follow that, if for a set of scalars $\{\alpha_k\}$, the series $$\sum_{k=1}^{\infty}\alpha_ke_k$$ converges to an $x \in H$, ...
0
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2answers
40 views

Derivative as a continuous mapping of a subset of $R^n$ into the set of all invertible elements

I have been studying the proof of the inverse function theorem in the Rudin's book (Principles of mathematical analysis). Near the end of the proof, he makes the following statement "(...) observe ...
1
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0answers
19 views

Ito's lemma in infinite dimensional spaces

I'm trying to use the ito's lemma in infinite dimensional spaces applicatte to $F(X)=\Vert AX\Vert^{2}$, where $A$ is a linear map. But i I have trouble calculating the integral $\int_{0}^{t}\langle ...
2
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1answer
26 views

Exchanging limit and evaluation of argmax

Let $g_n:\mathcal X \to \mathbb R$ for $n \in \mathbb N$ and $f:\mathcal X \to \mathbb R$. If $\lim_{n \to \infty} g_n(x) = f(x)$ for all $x \in \mathcal X$, then does the following hold? $$ \lim_{n ...
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0answers
20 views

linear dependence and measure theory

Let $(\Omega,\Sigma,\mu)$ be an measure space. Further let $E:=L^p(\Omega,\Sigma,\mu)$ where $\dim E \ge 2$. First I had to show, that if there are $[f],[g] \in E$, so that $\forall \alpha,\beta \in ...
1
vote
2answers
47 views

A real analysis problem on the integral inequality.

For fixed $ 0 < \alpha < \beta $, is there a positive constant $C_0$, depending only on $\alpha$ and $\beta$, such that for any bounded measurable function $ \varphi : \mathbb{R}^+\rightarrow [0 ...
1
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2answers
34 views

Are these examples of a normed space and a Banach space?

Our professor gave two examples of spaces of sequences one of which is Banach and the other not: Consider the space of sequences $X$, where only finitely many terms are non-zero, with the norm ...
2
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0answers
17 views

Orbits under small perturbations

Suppose $T$ is a bounded linear operator on $l_2$ and $x\in l_2$ is a vector such that the orbit $(T^{n}x)$ is linearly independent. Can one find an $\epsilon>0$ such that for all ...
0
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0answers
33 views

prove that K is Lp- bounded operator

Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space, $1\leq p <\infty$, and suppose that $k:X\times X\rightarrow \mathbb{F}$ is an $\Omega \times \Omega$ measurable function such that for $f$ ...
2
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1answer
29 views

Are $X'\otimes Y$ and $\mathfrak L(X,Y)$ isomorphic?

Let $\mathbb F\in\left\{\mathbb C,\mathbb R\right\}$ $X$ and $Y$ be normed $\mathbb F$-vector spaces $X'$ be the topological dual space of $X$ $\mathfrak L(X,Y)$ be the set of bounded, linear ...
0
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1answer
12 views

show that for any compact set $K \subset X$, there exist an integer $I\geq 1$ such that:

Let X be a banach space. show that for any compact set $K \subset X,\forall \epsilon > 0$ there exists an integer $I\geq 1$ and a collection of points $x_1, \dots ,x_I \in X$ such that: ...
2
votes
1answer
23 views

range of weighted shift operator

Consider the weighted shift operator on $\ell^2$ space defined by $T(x_0, x_1, x_2, ...) = (0, x_0, 2x_1, 3x_2, 4x_3, ...)$ with domain $$\mathcal{D}(T) = \{(x_n) \in \ell^2 : ...
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0answers
42 views

Is it possible to build a Banach space given a metric function space?

I have a curiosity, For a given function space $V$, and a metric $d(\cdot,\cdot)$ is it possible to construct a non-empty subspace $W$ such that $(W,d)$ is a Banach space?
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1answer
30 views

Pointwise evaluation of $L_2$ Fourier Transform

We know, that the $L_2$-Fourier Transform of a function $f\in L_2$ is defined as a limit of $L_2$ functions (e.g. $\ \mathcal{F} f=\lim_{n\to \infty} \int_{-n}^{n} f\cdot \chi_{(-n,n)}\ d\lambda ...
2
votes
0answers
35 views

$f \in L_2$ bandlimited implies $f$ equal to continous function a.e. (without using Parley-Wiener)

I was wondering, if my proof is right as I didn't find any similar statements in books or the internet without using the Parley-Wiener-Theorem: If we have $f \in L_2(\mathbb{R})$, bandlimited (i.e. ...
1
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0answers
40 views

Holomorphic semigroup can be extended to a strongly continous semigroup on $L^{p}$?

I have a question about $C_{0}$-semigroup theory. Let $(H,(\, , \,))$ be a real Hilber space and $(H_{\mathbb{C}}, (\, , \,))$ the its complexification. Any linear operator $(L,D(L))$ on $H$ can be ...
2
votes
1answer
57 views

$\frac{d}{dt}$ on $\mathbb{R}$ is not a Fredholm operator?

I encountered a statement in a book that $\frac{d}{dt} : L^2_1(\mathbb{R}) \longrightarrow L^2 (\mathbb{R})$ is not a Fredholm operator, where $L^2_1$ is the first Sobolev space of the $L^2$ space. ...
0
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1answer
25 views

Alaoglu & Krein-Milman to show bounded convex weak-$*$ closed subset has extreme point

Let $X$ be a normed linear space and $K$ a bounded convex weak-$*$ closed subset of $X^{*}$. I need to show that $K$ possesses an extreme point; however, I am not entirely sure how to do this. I ...
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0answers
22 views

Regularity properties of the inverse of Laplacian

I have been studying the so-called Helmholtz operator $(1-\partial_x^2)^{-1}$ using its kernel representation: $$ (1-\partial_x^2)^{-1}f(x)=\int_{\mathbb{R}}\frac{1}{2}e^{-|x-y|}f(y)dy $$ And a ...
4
votes
1answer
57 views

proof of an equality norm

Let the mapping $T:\ell^{2}\rightarrow \ell^{2}$ is defined as follow. $$T(x_1,x_2,\ldots,x_n,\ldots)=(x_1,\dfrac{1}{2}x_2,\ldots,\dfrac{1}{n}x_n,\ldots)$$ In this case, i've easily earned: ...
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2answers
19 views

Continuous function is measurable with respect to Borel sigma-algebra. [closed]

There are two measurable spaces with Borel sigma-algebras on them $(X, \mathcal B (X)) $ and $(Y, \mathcal B (Y)) $. There is also a continuous function $f:(X, \mathcal B (X)) \to (Y, \mathcal B (Y)) ...
3
votes
1answer
62 views

Why are these distributions positive?

I am trying to understand some calculations in a paper by Sidney Coleman. He is showing that certain distributions are positive. The paper can be found here. What I am talking about is happening at ...
0
votes
1answer
35 views

How to solve Fredholm Integral Equation of the Second Kind in $C[0,1]$

I need to solve, in $C[0,1]$, the equation $\displaystyle x(t) - \lambda \int_{0}^{\pi}(\sin t \cos s)x(s) ds = \sin t$. Adding the integral part to both sides, I obtain $x(t) = \sin t + \lambda ...
0
votes
1answer
36 views

Weak solution in $\mathbb{R}^{N}$

I'm bit confusing about definition of weak solution. If I have the following problem: $\begin{cases} \tag{P} -\Delta u = f \textrm{ in } \Omega, \\ u = 0 \textrm{ in } \partial\Omega, \end{cases}$ ...
1
vote
3answers
46 views

$f(x)$ and $xf(x)\in L^2(\mathbb{R})$ then $f(x)\in L^1(\mathbb{R})$

If $f(x)$ and $xf(x)\in L^2(\mathbb{R})$ then $f(x)\in L^1(\mathbb{R})$. I know that if $E$ is of finite measure, then we can infer from $f(x)\in L^2(E)$ to get $f(x)\in L^1(E)$. However, now ...
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0answers
16 views

Taylor series approximation of inverse trigonometric function

Suppose we have a function of three variables $a,b,c$ defined as, $f(x,y,z)=\arctan\left(\frac{\sqrt{x^2y^2-z^2}}{y^2-z}\right)$. Suppose $x=a, y=b, z=c$ satisfy the following property: (1) ...
1
vote
1answer
32 views

What is meant by $B(X,Y)$ and (how) are these theorems equivalent?

I was hoping someone could explain what is meant by $B(X,Y)$ in the following statement of the Banach-Steinhause theorem (this comes directly from Wikipedia): Theorem (Uniform Boundedness Principle) ...
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0answers
38 views

The benefit of writing Banach space theory in categorical language!

I was wondering if there exists a special benefit of writing Banach space theory in categorical language? I mean does there arise a hint of the existence of a connection with other mathematical field ...
4
votes
1answer
37 views

Sum of closed spaces is not closed

I recently encountered the theorem that the sum of a closed (linear) subspace with a finite dimensional subspace is closed subspace of the Banach space in which it is contained. However, this came ...
1
vote
1answer
50 views

$T$ is an $L^2$-bounded operator; find its norm

We have the integral operator $$ P:L^2(\Bbb R^n)\to\{\text{meas.functions}\;:\;\Bbb R^n\to\Bbb R\} $$ defined as $$ Tf(x):=\int_{\Bbb R^n}L(x,y)f(y)\,dy $$ where $L$ is a measurable function on $\Bbb ...
0
votes
1answer
72 views

Determine whether the differential operator is compact in the following cases

Given the differential operator $\displaystyle Tx(t)=\frac{dx}{dt}$, I need to determine (and be able to justify) whether it is compact in the following three cases: $T: C^{1}[0,1]\mapsto C[0,1]$ ...
2
votes
1answer
61 views

Space of test functions defined by norms

This is the problem assigned: So I know that a locally convex Hausdorff space is defined by a vector space and a family of seminorms. So is part $a$ just wanting me to show that $\|\phi\|_m$ is in ...
1
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1answer
30 views

Why is $tr((a_{ij})=\sum\limits_{i=1}^n a_{ii}$ k-positive for all $k$?

Let $A$ be a $C^*$-algebra and $$Tr:M_n(A)\to A,$$ $$(a_{ij})\mapsto \sum\limits_{i=1}^n a_{ii}.$$ The claim is that this map is k-positive for all $k\in\mathbb{N}$. Let $k\in\mathbb{N}$ and consider ...
1
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0answers
29 views

Nonconvex functional show inf is zero but inf is not attained

This is Exercise 8.22 from John Hunters Applied Analysis books. The question says: Consider the non-convex functional: Defined by Where W is the Sobolev space of functions that belong to ...
0
votes
1answer
25 views

unitary operator between two Hilbert subspaces

$H$ is a Hilbert space. $P, Q$ are projections. For every $x\in P(H)$, we have decomposition $x = Qx +Q^\perp x$. Then, can we find a unitary operator from the space generated by all $Qx$, $x\in ...
1
vote
1answer
38 views

If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism.

Let $X$ and $Y$ be metric spaces, $X$ compact, and $T:X \to Y$ bijective and continuous. Show that $T$ is a homeomorphism. My attempt: We need only show that $T^{-1}$ is continuous. Let $M ...
1
vote
1answer
41 views

Diagonal operators on infinite dimensional Hilbert spaces

the following is a short question regarding a theorem from a quantum mechanics book I am working through but the question is a mathematical one. There is a theorem which states: Theorem: The ...
1
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1answer
34 views

Is the space $\mathbb{R}_+\times S\times S$ linear?

The space $\mathbb{C}$ (or even $\mathbb{R}^2$), which is a linear space over $\mathbb{R}$, can be obtained from the Cartesian product $\mathbb{R}_+\times S$ by gluing to the point the layer $0 ...
0
votes
1answer
28 views

solving a complex integral (qft)

I am wondering how to evaluate this integral: $$I = -i\int_{-\infty}^{\infty}\textrm{d}p\dfrac{pe^{ipr}}{\sqrt{p^2+m^2}}$$ In my physics text book the substitution $p=i\rho$ is made together with ...
1
vote
1answer
19 views

$C(\overline\Omega)$ and $C_0(\Omega)$

Suppose $\Omega$ is a smooth open bounded domain in $\mathbb{R}^n$. Is $C_0(\Omega)$ a subset of $C(\overline\Omega)$? I think it is not because I can take $\Omega = (0,1)$ with $f(x) = \frac{1}{x}$ ...
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0answers
15 views

Is the generated semigroup by an elliptic operator be the transition semigroup?

I am considering the time homogeneous Ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dW_t,$$ where $b,\sigma$ currently are only assumed to be global Lipschitz for the existence of solution $X_t$. The ...
0
votes
0answers
35 views

Criteria for convex set compactness in weak*-topology.

I am trying to solve the following task: convex set $M$ in dual space $X^{*}$ is weakly*-compact if and only if it is closed and bounded. Let me show what I already happened to do about the task. ...
0
votes
1answer
17 views

Function integrability

I am considering the following function. $f(x)=\sum_{i=1}^d{\mid x_i\mid^2}$ for $x\in\mathbb{R}^d$. I am now considering the inverse function $g(x)=\frac{1}{f(x)}$. Claim: g is integrable if ...
0
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1answer
14 views

Relation between the distance from a point to the resolvent set of a self-adjoint operator and the norm of a related operator

$H$ is a Hilbert space and $B(H)$ is the Banach space of its bounded linear operator on $H$. Assume $A\in$ $B(H)$, $A$ is self-adjoint. Suppose $\lambda_{0}$ is in the resolvent set of $A$. Show that: ...
1
vote
1answer
19 views

Calculating the norm of $T(x)=(f_1(x),f_2(x),\dots)$

Let X be a Banach space and $f_i \in X^*, i \in \mathbb{N}$ such that $$ \sum_{i=1}^\infty|| f_i(x)|| < \infty, \forall x \in X. \hspace{2cm}(I) $$ Calculate the norm of $T: X \to ...
4
votes
1answer
54 views

This linear operator has no eigenvalues

Let $T : L^2(\mathbb R) \to L^2(\mathbb R)$ be a linear operator defined by $$(Tf)(x)=f(x+1).$$ Show that $T$ has no eigenvalues, i.e., there exists no $f \not= 0$ in $L^2(\mathbb R)$ such that ...
2
votes
2answers
40 views

Norms with complex numbers over Hilbert Spaces

Let $H$ be a Hilbert space and $v,w \in H$ ans a be a scalar. Prove that $\|v\| \leq \|v+aw\|$ for all scalar a iff (v,w)=0 for real and complex cases. I want to choose a such that $\bar{a}(v,w)$ ...
-1
votes
2answers
43 views

Cardinality of the family of all closed subspaces of a separable Banach space

Is it true that the cardinality of the family of all closed subspaces of a separable Banach space is less than or equal to continuum ? (or, is countably infinite?) Thanks for any answer, comment or ...
1
vote
1answer
44 views

Applaying equivalence of norms to show a sequence is a Cauchy sequence

Let $\|\cdot\|$ be any norm on $\mathbb R^n$ prove that a sequence $x \in \mathbb R^n$ is a Cauchy sequence under $\|\cdot\|_2$ if and only if it is a Cauchy sequence under any $\|\cdot\|$. I tried ...