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, ...

learn more… | top users | synonyms (1)

2
votes
1answer
51 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
votes
1answer
42 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.)
0
votes
1answer
23 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 ...
1
vote
0answers
52 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
votes
2answers
76 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 $X=\bigcup X^n$, ...
1
vote
0answers
29 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 ...
6
votes
0answers
196 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, $f$ ...
0
votes
1answer
62 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 ...
1
vote
1answer
48 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 ...
2
votes
1answer
35 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
22 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
39 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
19 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
votes
0answers
23 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 ...
2
votes
0answers
23 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 ...
0
votes
1answer
32 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 ...
1
vote
1answer
31 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 ...
2
votes
2answers
55 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
29 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 ...
0
votes
1answer
28 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.
4
votes
0answers
27 views

Differentiation in Besov-Zigmund 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 ...
1
vote
0answers
30 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 ...
1
vote
1answer
33 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 ...
4
votes
4answers
92 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 ...
2
votes
1answer
44 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 ...
1
vote
1answer
25 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 ...
0
votes
0answers
17 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 ...
0
votes
0answers
35 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
44 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
96 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
46 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
55 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 $S= (S^{\perp})^{\perp}$?
2
votes
1answer
35 views

How to calculate this functional derivative?

How can I calculate the functional derivative of this functional? $$F[x](t) = \int_{0}^{t}x(t_1)a(t_1)\left \{ \int_{0}^{t_1}x(t_2)b(t_2) \,dt_2\right \} dt_1 .$$ Where $a(t)$ and $b(t)$ are real ...
2
votes
0answers
31 views

Borel measure and positive linear forms

I'm just starting to learn about positive linear forms. If we call $C_{C}(X)$ the space of all continuous functions with compact support from domain $X$ and $\mathbb{C}$ (with $X$ a locally compact ...
2
votes
0answers
23 views

If $X$ is a LCHS and $f \in C_{C}(X)$ and $\mu$ is a Borel measure, then $f \in L^{1}(d\mu)$.

I want to prove the following statement: If $X$ is a locally compact Hausdorff topological space, and $f \in C_{C}(X)$ ($f$ is a continuous function with compact support), and if $\mu$ is a Borel ...
0
votes
0answers
31 views

When is the second derivative of a $H^{1}$ function in $L^{2}$?

Is there a characterisation of all functions $\phi\in H^{1}$ such that for given functions$\{g^{ij}\in L^{\infty}\}$ then $\sum_{ij} g^{ij}\phi,_{ij}\in L^{2}$ where $\phi,_{ij}=\frac{\partial ...
5
votes
2answers
117 views

There is no norm in $C^\infty ([a,b])$, which makes it a Banach space.

Does anyone knows a reference, which proves the following: Let $a,b\in \mathbb{R}$ with $a<b$. There is no norm in the space $C^\infty([a,b])$, which makes it a Banach space.
1
vote
1answer
29 views

If $T$ is topologically transitive and $X$ is separable and complete then there exists a dense set of points with dense backward orbits.

I am trying to solve exercise 1.2.7 from Grosse-Erdmann and Peris' book Linear Chaos. It is stated as follows: Let $T:X\rightarrow X$ be continuous on a separable and complete metric space $X$ ...
0
votes
0answers
18 views

About the compactness condition in Schauder fixed point theorem

The theorem is Let $X$ be a locally convex topological vector space, and let $K ⊂ X$ be a non-empty, compact, and convex set. Then given any continuous mapping $f: K → K$ there exists $x ∈ K$ ...
1
vote
1answer
34 views

$T\in\mathcal{L}(X,Y)$ maps closed bounded subsets onto closed subsets $\implies$ Range $T$ is closed.

Given two normed spaces $X$ and $Y$ and let $T$ be a bounded linear operator $T:X\to Y$. Assume that $T$ maps bounded and closed subsets of $X$ onto closed subsets of $Y$. Show that the range of $T$ ...
1
vote
1answer
33 views

Which functions lies in $H^{loc}_{s}\setminus H_{s}$?

We put $H^{s}=$The Sobolev spaces, and $H^{loc}_{s}=$The localized Sobolev spaces. We note that, $H_{s}\subset H^{loc}_{s};$ also this. Bit roughly speaking, I am interested in knowing that how big ...
0
votes
1answer
49 views

Open sets are not relatively compact

The following is a question about the answer given here: I have been trying to prove that if $X$ is an infinite dimensional Banach space and $O\subseteq X$ is an open set such that its closure ...
1
vote
1answer
52 views

Proving $u$ is compact whenever $u^\ast$ is

Let $X,Y$ be Banach spaces and let $u: X \to Y$ be a linear operator. Let $u^\ast: Y^\ast \to X^\ast$ denote its transpose and assume that $u^\ast$ is compact. I am trying to prove that $u$ is ...
0
votes
1answer
24 views

A question about the definition of $(X\rtimes\Gamma)$-C*-algebra

Here is a quotation in the book "C*-algebras and Finite-Dimensional Approximations": Instead of considering the *-algebra of finitely supported functions from $\Gamma$ to $C(X)$ (C(X) denotes all ...
5
votes
1answer
69 views

Is $C^{\infty}([0,1])$ a Banach space?

I have read that the answer is no, but I am unable to prove it. Give $C^{\infty}([0,1])$ the metric $$d(f,g) = \sum_{j=0}^{\infty} 2^{-j} \frac{||(f-g)^{(j)}||}{1 + ||(f-g)^{(j)}||}$$ associated to ...
0
votes
0answers
32 views

Question about the zeros of $\zeta_{H}(s,a) \pm \zeta_{H}(s,1-a)$.

Assume $\zeta_{H}(s,a)$ is the Hurwitz Zeta function. Note that for $a=\frac13,\frac14,\frac16$ the zeros of: $$\zeta_{H}(s,a) + \zeta_{H}(s,1-a)$$ are the same as the non-trivial zeros $\rho$ of ...
0
votes
2answers
49 views

Extending Positive Functionals: Linearity

How does regularity provide linearity? Given the full Banach space of bounded functions over a suitable set: $$\mathcal{B}:=\{f:\Omega\to \mathbb{C}:\|f\|_\Omega<\infty\}$$ and a linear subspace ...
0
votes
0answers
12 views

From essential oscillation to a continuous representative

Let $u$ be a measurable function such that for every(former: a.e.) $x\in \Omega$ there holds for sequences $R_n,\delta_n\to 0$ that$$\omega_n:=ess-osc_{B_{R_n}(x)} u\leq \delta_n. \tag{1}$$ Edit: ...
6
votes
1answer
54 views

Compactness of the Volterra opelator

The Volterra operator is given as \begin{eqnarray} (Vf)(x)=\int_0^xK(x,y)f(y)\,{\rm d}y. \end{eqnarray} By the Arzelà–Ascoli theorem, $V\colon C^0[0,1]\rightarrow C^0[0,1]$ is compact operator. But, ...
0
votes
1answer
20 views

Suppose $a$ and $b$ are positive numbers. Prove that for every $\epsilon>0$, $\exists$ $C(\epsilon)$ such that $ab\leq\epsilon a^p+C(\epsilon)b^q$.

Suppose $a$ and $b$ are positive numbers. Prove that for every $\epsilon>0$, there is a constant $C(\epsilon)$ such that $ab\leq\epsilon a^p+C(\epsilon)b^q$ where $\frac{1}{p}+\frac{1}{q}=1$. I'm ...