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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Is $f(x) \in L^p(\mathbb R)$ always bounded for $x\longrightarrow\pm\infty$?

I need to prove the following result on the derivative of an Hilbert transform for $f,f'\in L^p(\mathbb R)$ $$\mathcal H\bigg\{\frac{df(x)}{dx}\bigg\}=\frac{d}{dx}\mathcal Hf(x) $$ In particular ...
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1answer
31 views

Continuous Quadratic Form $\implies$ Continuous Sesquilinear Form

Given a Hilbert space $\mathcal{H}$. Consider a quadratic form $q:\mathcal{H}\to\mathbb{C}$. Define its inducing sesquilinear form: $$s:\mathcal{H}\times\mathcal{H}\to\mathbb{C}: ...
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1answer
56 views

Are all definite integrals considered functionals?

In my Optimization class, we are messing around with some Calculus of Variations in an effort to find functions which minimize functionals. In these cases, the spaces we're working with are spaces ...
2
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0answers
18 views

boundedness of ball-like convex subsets in $\mathbb{R}^n$

Let $K \subset \mathbb{R}^n$ be a subset with the following three properties: (i) $K$ is convex (ii) $K$ is symmetric about $0$, that is if $k \in K$, then $-k \in K$ as well. (iii) If $l$ is any ...
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1answer
42 views

Banach Algebras: Continuity of Inversion?

Context: This question is related to this thread: Spaces of Functions Given a topological space $X$ and a Banach algebra with unit $B$. Consider a continuous map $F:X\to B$ that is invertible ...
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1answer
33 views

Is $\sum c_n z^n$ analytic when $c_n$ is Banach-valued?

I'm trying to define "Analytic function". I want a definition that covers all interesting cases. To be specific, let me explain what exactly I want Here is the definition of analytic function in ...
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1answer
40 views

Infinite-dimensional version of Gram matrix is invertible

We all know that a Gram matrix (a matrix with entries that are inner products of basis functions) is a invertible. Suppose I have $a_{ij} = (h_i, h_j)_H$ where the $h_j$ are basis functions of a ...
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1answer
71 views

For which exponents $\gamma$, the function $|x|^{1/2}$ is $\gamma$-Holder continuous?

I have to prove the following. Let $$u(x):=|x|^{1/2}$$ if $$|x|\le 1$$ For which exponents $\gamma\in (0,1]$, $u\in C^{0,\gamma}([-1,1]).$ The answer should be $\gamma\in (0,\frac{1}{2}]$, but I ...
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1answer
48 views

Isomorphism between Euclidean space and its conjugate

I know that, if $H$ is a Hilbert space, for any continuous linear functional $f\in H^{\ast}$ there is a unique element $x_0\in H$ such that $\forall x\in H\quad f(x)=\langle x,x_0\rangle$. Moreover, ...
0
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1answer
40 views

Local and global minimum/maximum and continuity

My task was to determine the global minimum of a function $f(x,y) = x^3 + y^3 - 3xy$ on the square $[0,2] \times [0,2]$. I first calculated the points where the gradient $(grad f)(x,y) = (3x^2 - 3y, ...
2
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2answers
84 views

$\phi_n \rightarrow \phi$ weakly-$*$, then $\|\phi\|\leq \limsup_n \|\phi_n\|$.

$\phi_n \rightarrow \phi$ weakly-$*$, then $\|\phi\|\leq \limsup_n \|\phi_n\|$. My attempt: $$\|\phi\| = \sup_{\|x\| = 1} |\phi(x)| \leq \sup_{\|x\| = 1} \lim_n |\phi_n(x)|$$ Using an ...
2
votes
1answer
90 views

show that every compact operator on Banach space is a norm-limit of finite rank operators (in a particular way, under the given hypotheses)

Let $X$ be a Banach space and suppose there is a net $\{F_i\}$ of finite-rank operators on $X$ such that $(a)$ $\sup_i||F_i||<\infty$ ; $(b)$ $||F_ix-x||\to 0$ for all $x$ in $X$. Show that if $A$ ...
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0answers
37 views

Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...
2
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0answers
50 views

Reference request for the proof of the Brodskii–Milman fixed point theroem for isometries

Can any one help me to access the paper M.S Brodskii and D.P Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948) 837–840 in Russian? or to prove the theorem If $K$ is a ...
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1answer
32 views

Show that $\{x_1,…,x_n\}$, where $x_j=t^j$, is a linearly independent set in the space $C[a,b]$.

Show that $\{x_1,...,x_n\}$, where $x_j=t^j$, is a linearly independent set in the space $C[a,b]$. I think I can use properties of polynomials in $R[x]$ here, but I'm not sure. Using $\sum_{i=1}^n ...
3
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0answers
60 views

Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form $$ \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} $$ It is well known that if $f$ is ...
0
votes
1answer
80 views

Dirac delta distribution & integration against locally integrable function

I was reading the a lecture note online about distribution theory and it said: The Dirac delta distribution $\delta \in D'$ is defined as $\delta(\varphi)= \varphi(0) $, and there's no locally ...
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1answer
55 views

Maximal chain in the collection of all invariant subspaces for compact operator $K$

Let $X$ be a Banach space over ${\Bbb C}$, and $K\in K(X)$ ($K(X) = $ compact operators space). Show that if ${\cal L}$ is a maximal chain in the $Lat K$ ($Lat K = $ the collection of all invariant ...
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1answer
65 views

Operators on non-separable Banach spaces have non-trivial invariant subspaces

Show that if $T\in B(X)$ and $X$ is not separable, then $T$ has a nontrivial invariant subspace. I know that $\ker (T)$ and $\operatorname{ran}(T)$ are invariant $T$-subspace. So if $\ker T\neq ...
2
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1answer
68 views

Existence of a dense hyperspace

How do you prove that every infinite dimensional normed space contains a dense hyperspace? (Where hyperspace is defined to be maximal proper subspace.)
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1answer
34 views

Prove that solution of a variational problem exists

Let $X$ is a Hilbert Spaces We define two operators $$a:X\times X\rightarrow\mathbb R$$ and $$b:X\rightarrow\mathbb R$$ where $a$ is a symmetric, bounded, strongly positive operator, and $b$ is a ...
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0answers
66 views

How to prove the uniqueness of a specific root?

Let us define: $$F(x):=\int_0^Tf(t)\cos(x\,t)dt-\frac{\sin(T_0\,x)}{T_0\,x}$$ where: 1). $0<T_0<T \in\mathbb{R}^+$ are both positive real constants, and 2). $0\leqslant f(x)\in C^{\infty}$ ...
3
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2answers
96 views

Any relations between the weak topology on a Banach Space and the weak topology on CW complexes?

I'm learning about CW complexes, which we'll say are topological spaces $X$ that admit a filtration $\emptyset \subset X^0 \subset X^1 \subset \cdots \subset X^n \subset \cdots$, with $X=\bigcup X^n$, ...
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0answers
86 views

functional analysis: show L^1 integral operator has norm 1

I just started my course in functional analysis and have already stumbled across some things I don't understand, which are quite basic :(. In my lecture notes it says: Let $\mu$ be a measure on a ...
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0answers
234 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
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1answer
81 views

Concepts of isomorphisms of linear spaces with a norm and inner product

If I have a topological space, I say that a homeomorphic map preserves the structure of this space. Thus, in order to preserve topological properties we want to have a continuous bijection with a ...
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1answer
52 views

The space $ \left( \sum \ell_p^n \right)_2$ is reflexive.

Let $\ell_p^n:= (\mathbb R^n, \|\cdot\|_p)$. I want to show that the space $$ \left(\sum_{n=1}^\infty\ell_p^n \right)_2 := \left(\left\{ (x_n)_{n \in \mathbb N} : x_n \in \ell_p^n ...
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1answer
36 views

Is the linear functional that sums the coefficients of $p$ continuous on $P([a,b])$ with $\|\cdot\|_\infty$.

Is the linear functional that sums the coefficients of $p$ continuous on $P([a,b])$ with $\|\cdot\|_\infty$. My attempt: Claim: It is continuous. When $1\in [a,b]$: Since a linear functional is ...
0
votes
1answer
25 views

Linear transform $T$ such that $T(b^x)=b(b-1)^x$

The title pretty much says it all. I'm trying to find a linear transform, maybe a vague analog of a derivative, that has the property that if $f(x)=ab^x$, then $T(f)=ab(b-1)^x$, analogous to the ...
2
votes
1answer
53 views

Eigenfunctions and spectrum of $T:H \to H^*$ where $H$ is a Hilbert space

Let $H$ be a Hilbert space with dual $H^*$. Suppose $T:H \to H^*$ is a linear bounded symmetric operator. (We probably don't want to identify $H$ with $H^*$). Can we talk about the ...
0
votes
1answer
71 views

Weak derivatives equals zero

Im just learning sobovel space, I was wondering if the weak derivate holds similar things of the original derivate. Let $U\subset \mathbb{R^n}$ is a open set, and $u\in W^{1,p}$ if $$Du=0 \ \ a.e$$ ...
2
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0answers
31 views

The existence of integral equations solution for a 2-dimensional unknown function

Suppose $f(\cdot,\cdot)\in C[0,1]^2$ is a kernel. $f$ is integratble $\int_0^1\int_0^1 f(x,y)dxdy<\infty$. $a,b\in C[0,1]$ are known functions, and $z(\cdot,\cdot)\in C[0,1]^2$ is a 2-dimensional ...
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0answers
29 views

Invariant subspaces and their complements

The following is Exercise 6.4.2 of Conway's Functional Analysis: Suppose $T\in B(X)$ where $X$ is a Banach space. Prove that $M$ is invariant under $T$ if and only if $M^{\perp}$ is invariant under ...
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1answer
36 views

Showing that a map $x \to \|x\|$ is continuous?

I am given this: Consider a real Banach space $X$ with norm $\|*\|$. 1) Show that the map $x\to \|x\|$ from $X$ to $\mathbb{R}$ is continuous. Is it uniformly continuous? 2) Show that the maps ...
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1answer
66 views

Can essentially bounded function take infinite value on measure zero set?

I know that $\|f\|_{\infty}=esssup_{x\in X}(f(x))$, which means we can neglect measure zero sets in our definition of essential supremum. I am comportable when the function is bounded on all points ...
3
votes
2answers
112 views

Showing that $C^1[0,1]$ is a Banach space with the $||f||=||f||_\infty + ||f^\prime||_\infty$ norm.

So I am a bit stuck on where to begin with this one... Show that $C^1[0,1]$ with the norm defined as $||f||=||f||_\infty + ||f^\prime||_\infty$ is a Banach space. I started with an arbitrary cauchy ...
0
votes
0answers
38 views

Finding inverse of a general linear transform

I'm not a mathematician, so I may abuse some notation here. Please comment for any clarification. Let's define a general linear transform as $$\int_XK(\mathbf{\omega},x)f(x)dx$$ where $X$ is some ...
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1answer
59 views

Sobolev embedding counterexample

I trying to find a counterexample to show that $$W^{1,p}(\mathbb{R ^n}) \nsubseteq C^{0,\alpha}(\mathbb{R^n}) $$ for $p>n$ and $\alpha > 1 -\frac{n}{p}$. No clue yet, thanks for your help.
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1answer
81 views

Differentiation in Besov–Zygmund spaces

This is my second question in a short time on Besov spaces. I apologize. I am having a rough time with them and I really need to understand this spaces quickly. The Besov spaces ...
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0answers
31 views

Eigenvalues that are functions

Let us have the Laplacian on a compact manifold $M$. Suppose I have some equation of the form $$-\Delta u(x) = f(x)u(x).$$ If $f \equiv c$ were a constant, this would be an eigenvalue problem ...
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1answer
34 views

$A,B$ bounded $\Rightarrow$ $A+B$ bounded

I have read that if the subsets $A$ and $B$ of a topological vector space are bounded, i.e. for any neighbourhood $U$ of $0$ there is an $n>0$ such that, for all $|\lambda|\geq n$, $M\subset\lambda ...
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votes
4answers
132 views

What does a norm $\|x\|$ goes to infinity mean?

I am looking into Coercive functions. The definition says : A continuous function $f : \mathbb{R}^n → \mathbb{R}$ is called coercive if $$\lim_{\|x\| \to \infty} f(x) = + \infty$$ What does a norm ...
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1answer
53 views

Compactness of an operator on $c_0$ in terms of its infinite matrix representation

Let $A\in {\cal B}(c_0)$ (${\cal B}(c_0)$ is linear bounded operators on $c_0$) and for $n\geq 1$, define $e_n \in c_0$ by $e_n(m)=\delta_{nm}$. Put $\alpha_{nm}=(Ae_n)(m)$ for $n,m\geq 1$. we have $M ...
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1answer
63 views

Smoothing effect for weak solutions of heat equation

Let $u_0 \in L^2$ and $f \in L^2(0,T;H^{-1})$ and consider the solution $u \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ of $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some BC (eg. zero Dirichlet). I am ...
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votes
0answers
50 views

Kernel of linear operator closed if domain non-$T_2$?

I read on my functional analysis text that the kernel of a linear operator $A:V\to W$ between two topological linear spaces is closed. My book don't require topological linear spaces to be Hausdorff ...
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0answers
47 views

Exercise of direct sum of operators: could someone please check my work

I tried to do this exercise and was wondering of someone could please read my work and tell me if it is correct: Let $u: X \to Y$ and $u': X' \to Y'$ be bounded linear operators between Banach ...
2
votes
1answer
54 views

why say “$\mathbb{R}$-tree”?

Definition. An $\mathbb{R}$-tree is a metric space $(X,d)$ such that there is a unique geodesic segment (denoted $[x,y]$) joining each pair of points $x,y \in X$; if $[x,y] \cap [y,z] = ...
1
vote
3answers
105 views

Showing $T: X\rightarrow Y$ is a linear map, is one-to-one… Over-thinking question?

so my question is as follows: Suppose that $X$ and $Y$ are normed linear spaces and that $T: X\rightarrow Y$ is a linear map (ie $T(\alpha x_1+\beta x_2) = \alpha T(x_1) + \beta T(x_2) \forall ...
1
vote
1answer
75 views

spectral decomposition of a bivariate function

Now I have a function $f=f(x,y)$, smooth and symmetric(i.e. $f(x,y)=f(y,x)$ everywhere), with arguments defined on a compact set: $(x,y)\in[0,1]\times[0,1]$. I'd wish to know if $f$ can be expanded ...
0
votes
1answer
132 views

Orthogonal complement of vector spaces

Let $V$ be a vector space. Here I do not restrict $V$ to be finite dimensional. Let $S$ be a vector subspace of $V$. Why is $S\subset (S^{\perp})^{\perp}$ rather than $S= (S^{\perp})^{\perp}$?