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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7
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3answers
87 views

What does it mean for a topology on a mapping space to correspond to a type of convergence?

I've been asked to prove that three different topologies on $Y^X = \{f : X \to Y | f \text{ is continuous} \}$ correspond to three different types of convergence, but I don't understand exactly what ...
1
vote
0answers
11 views

Strong derivative of a compound map

I find the strong Fréchet derivative of $\Phi(h,\psi(h))$, where $\Phi:T_0\times T_\xi\to Y$ with $T_0, T_\xi, Y$ Banach spaces and $\psi:T_0\to T_\xi$ is strongly differentiable in $0$, evaluated in ...
0
votes
1answer
19 views

Closed restriction of an unbounded self-adjoint operator

Suppose $(A\,;\mathcal{D}(A))$ is an unbounded self-adjoint linear operator (obviously, $\mathcal{D}(A)$ must be dense) on a Hilbert space $\mathcal{H}$. Suppose $\mathcal{D}(C)$ is a proper dense ...
0
votes
0answers
8 views

Normalization for argument of maximum function

is it possible to normalize the maximum function of a certain argument ? Means: Is that $\theta_{ML} = arg \max\limits_{\theta} \{ \sum \limits_{n=1}^{N} |w_n w^*_{n+N}| - \Big( \frac{SINR + ...
0
votes
1answer
42 views

Convergence of a sequence in $l_2$

I am wanting to disprove (show that it is not the case) that for a sequence ${x_n} = x^n$, ($n\in\mathbb N$), that if $(x_i)^n \to x_i$ in $\mathbb R$,then $x^n\to x$ in $\ell_2$. I have gotten as far ...
0
votes
1answer
32 views

Solution to functional equation and Cauchy

I want to show that the only continuos solutions to the functional equation $g(x)+g(y)=g(\sqrt{x^2+y^2})$ is $g(x)=cx^2$ where c is a constant i.e. c=g(1). I think that I maybe could rewrite the ...
0
votes
2answers
43 views

A reflexive Banach space is separable iff its dual is separable

Let $(X,||\cdot||)$ be a reflexive Banach space. Prove that $X$ is separable if and only if $X'$ (the dual space of $X$) is separable. Does anyone have a hint for me? I have no idea where to begin
2
votes
0answers
26 views

Approximation of $L^2$ function by smooth functions on a manifold

Let $M$ be a $C^2$ compact Riemannian manifold with boundary. Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for ...
-4
votes
2answers
52 views

Is this function defined?

Let a function Let a function $g(f)= \parallel \bigtriangledown f\parallel / sin \parallel f \parallel $ Is $g $ defined for $\left \| f \right \| \leq $ 1? $\left| \left|. \right|\right|$ ...
3
votes
1answer
52 views

Approximating $L^p$ functions using Schwartz functions with compact support on the Fourier side

For $1\leq p<\infty$, how would you show for any $f\in L^p(\mathbb{R})$ and given $\epsilon>0$, there exists $L<\infty$ and $g\in \mathcal{S}(\mathbb{R})$ such that $\|f-g\|_p<\epsilon$ ...
0
votes
1answer
27 views

Lower bound and upper bound functions

I am studying for a test and really need to know some examples of function with upper bound and lower bounds. I hope someone can be kind enough to help.Thank you Please give examples of function ...
1
vote
1answer
23 views

Showing $f$ is an $L^p$ function if $f$ is "self-convoluted.

If $f$ is $L^2(\mathbb{R})$ and $f=f*f$, show that $f$ is $L^p$ for $2\leq p\leq \infty$.
1
vote
1answer
73 views

What topics have complex analysis among their prerequisites?

I have one spot left in my bachelor's curriculum and am trying to decide between complex and functional analysis. What the latter is good for, is more or less clear to me: e.g. for advanced ...
2
votes
0answers
33 views

Boundary conditions Legendre equation

I have Legendre's equation $$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$ Now I know that after substituting $\cos(\theta) =x$ we get a ...
2
votes
1answer
35 views

Breaking a Function in $L^{\infty}[0,1]$

Let $f\in L^{\infty}[0,1]$ s.t. $\|f\|_{\infty}=1$ $E:=\{x\in[0,1]:|f(x)|<1\}$ If $m(E)>0$, then is it possible to find $g,h\in L^{\infty}[0,1]$ such that ...
0
votes
0answers
20 views

Is $P_n$ closed in $L^2[(0,1)]$?

If $P_n$ is the set of polynomial functions of degree less than or equal to $n$, then is it closed in $L^2([0,1])?$ How would we go about showing this? Thanks!
0
votes
0answers
23 views

Sufficient conditions for weak continuity

Are there any "easily verifiable" sufficient conditions for weak (equivalently, weak*) continuity of (not necessailry linear) maps on the unit ball of $\ell^2$, mapping into $\ell^2$? Apologies for ...
1
vote
1answer
28 views

Weakly convergent sequence of operators

Let $H$ be a Hilbert space and $T_n\in\mathcal{L}(H)$. Suppose that for every $x\in H$ we have that $T_nx\rightharpoonup Tx$ where "$\rightharpoonup$" denotes weak convergence. Prove that ...
1
vote
0answers
25 views

Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
1
vote
1answer
16 views

Subspace of a Hilbert space with a distinct inner product

I don't really know where to begin with the following question: Let $ (H_0, \langle \cdot \rangle_0)$ be a closed subspace of $ (H, \langle \cdot \rangle )$ such that norms induced by $ \langle \cdot ...
0
votes
1answer
20 views

Proving an orthogonal projection of the Hilbert adjoint is just the adjoint

I'm facing the following problem: let $ H_0 \subset H $ be a $ T$-invariant closed subspace of Hilbert space $ H $ (i.e. $ T(H_0) \subset H_0 $) and $ P$ - an orthogonal projection of $ H $ onto $ ...
2
votes
1answer
38 views

Uniform convexity of equivalent intersection norm

I have two uniformly convex Banach spaces $E$ and $F$ (which are continuously embedded into a topological vector space $X$) whose intersection $E \cap F$ is non-trivial and is equipped with the norm ...
1
vote
1answer
27 views

Question about an exercise in Brezis' Functional analysis

In Brezis' Functional analysis, Sobolev spaces and partial differential eqautions, exercise 6.24 (3) asks to prove that for a self-adjoint operator $T\in \mathcal{L}(H)$, $H$ a Hilbert space, the ...
0
votes
0answers
18 views

Weak time derivative for functions $u \in L^2(0,T;L^2(\Omega))$

The weak time derivative of a function $u \in L^2(0,T;H^1)$ is defined to be $u' \in L^2(0,T;H^{-1})$ satisfying $$\int_0^T \int_\Omega u(t) \varphi'(t) = -\int_0^T \langle u'(t), \varphi(t) \rangle$$ ...
1
vote
0answers
21 views

spectrum of invertible operator

Let $T$ be a bounded linear operator and it is invertible. If we define $M(T)=\displaystyle\sup_{\left\|f\right\|=1}\left\langle ...
3
votes
1answer
17 views

Convergence of functions with different domain

Question: Is there a concept of convergence for functions $f_n: D_n \rightarrow X$ with different domains to a function $f: D \rightarrow X$? I know concepts like uniform convergence or almost ...
0
votes
0answers
13 views

Do these “algebraically well behaved” Function Spaces, exist?

Do there exist any Sets of Functions which are some combination of: Algebraically Closed: In the sense of Algebraic Functions. Differentially Closed: In the sense of Differentially Closed Fields. ...
0
votes
1answer
16 views

If $E$, $\overline{E}$ are orthogonal projections such that $\mathrm{range}(\overline{E})=\overline{\mathrm{range}(E)}$, then is $E\ge\overline{E}$?

I feel like this should be true. Let $\mathrm{range}(E)=A$ and $u$ be an arbitrary vector in a Hilbert space $H$, it is sufficient to show $\langle (E-\overline{E})u,u\rangle=0$. By Cauchy-Schwartz: ...
1
vote
2answers
100 views

Square root of unbounded operator

Let $T: \operatorname{dom}(T) \subset H \rightarrow H$ be a positive self-adjoint unbounded operator, then I want to define a UNIQUE(!) operator $A$ such that $A^{*}A = T$. Actually, this construction ...
0
votes
1answer
39 views

Precise meaning/implications of “a random variable belongs to a space” almost surely.

As far as I understood, by saying a random variable/vector $X$ belongs to a space $S$ (or takes values in $S$), one means that the measurable function $X$ is $S$-valued: \begin{equation} ...
2
votes
1answer
38 views

Square root of a Fourier series

This problem came to mind in conjunction with two earlier ones [1] [2]. Let $f(x)$ be positive square-integrable function on $[0,2\pi]$ with Fourier series $\sum\limits_{n=-\infty}^{\infty} ...
5
votes
1answer
64 views

Convergence of sequence of $L^{p}$ function

Given that $\Omega \subset \mathbb{R}^{n}$ is bounded. If you are given that $u_{k} \rightarrow u$ in $L^{p- \epsilon}(\Omega)$ and a functions $f: \mathbb{R} \rightarrow \mathbb{R}$ where ...
2
votes
0answers
60 views

Continuity of implicit function

I find a proof of the following theorem in A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа (pp. 492-493 here): Let $X,Y,Z$ be Banach spaces, $U$ a neighbourhood ...
1
vote
1answer
53 views

Continuity of an operator

I need some help with the following problem Let $X$ be a Banach space and let $T:X\rightarrow X$ be a linear map such that $T^2=T$. If the kernel of $T$ and the range of $T$ are closed, prove that ...
2
votes
1answer
32 views

Equivalent norms and inner product

It is not hard to give examples of normed spaces which are not inner product spaces. Now let $(V, \|\cdot\|)$ be an inner product space. Is it always possible to construct an inner product on $V$ ...
3
votes
1answer
36 views

Learning functional analysis and measure theory

I have taken a first course in real analysis and I'm currently studying analysis in $\mathbb{R}^N$ on my own. I want to start functional analysis after this, and I also want to learn measure theory ...
0
votes
1answer
29 views

Finite rank approximation of bounded operators on Hilbert space

Let H be a (finite dimensional) Hilbert space. The approximation property states that every bounded operator from H to itself can be approximated by a sequence of finite rank operators. My question ...
0
votes
1answer
12 views

About the von Neumann decomposition

The von Neumann theorem states that for any symmetric operator $f$, the domain $D_{f^\dagger}$ of its adjoint $f^\dagger$ is the direct sum of the three subspaces $D_{\bar{f}}$, $\aleph_z$, and ...
0
votes
1answer
40 views

Is the Product of Banach Spaces a Banach Space?

Let $X$ and $Y$ be two Banach spaces (not necessarily possessing the same norm). The product space $X×Y=Z$ is given the max norm, i.e. $\max(\Vert x\Vert, \Vert y\Vert)$, where $x$ is given the norm ...
2
votes
0answers
22 views

Hahn-Banach proof by extension of basis

Hahn Banach Theorem states that given a linear continuous functional $f$ on a subspace $N$ of a norm space $M$, it can be extended to a linear functional $F$ on the whole space $M$ and the norm of the ...
1
vote
0answers
32 views

Spectrum of adjoint operator

Let $X$ be a hilbert space and $T\in L(X)$ Show that: (i) $\sigma_c(T^*)=(\sigma(T))^*$ (ii) $\sigma_r(T)=((\sigma_p(T^*))^*)$\ $\sigma_p(T)$ (i): $"\subset"$ Let $\lambda\in\sigma_c(T^*)$ then ...
0
votes
1answer
37 views

Given $f:\mathbb{R}\to\mathbb{R}$ and $\forall\,x\in\mathbb{R}\,\,|f'(x)|\leq 1$

Given $f:\mathbb{R}\to\mathbb{R}$ and $\forall\,x\in\mathbb{R}\,\,|f'(x)|\leq 1$ ($f$ is differentiable on $\mathbb{R}$) Then, $\forall\,x\neq y\,\,|f(x)-f(y)|\leq|x-y|$ (Neither a contraction nor a ...
4
votes
1answer
36 views

A property of sobolev spaces

Let $W^{k,p}(\Omega):=\{y\in L^p(\Omega) : D^{\alpha}y\in L^p(\Omega)$ for all $|\alpha|\leq k\}$ I want to prove now that: (1) $u \in W^{1,2}(\mathbb R)$ is equivalent to (2) $u \in L^2(\mathbb ...
0
votes
0answers
11 views

operator ranges of a closed linear operator

I have a closed linear operator T on a Hilbert space H. Also I have that range R(T^n) is closed for some n $\in$ N and for non-zero $\lambda$, R(T-$\lambda$I) $\cap$ R(T^n) is closed and ...
1
vote
1answer
32 views

Let $X$ and $Y$ be normed spaces and $X$ compact. If $T: X\to Y$ is a bijective closed linear operator, show that $T^{-1}$ is bounded.

Let $X$ and $Y$ be normed spaces and $X$ compact. If $T: X\to Y$ is a bijective closed linear operator, show that $T^{-1}$ is bounded. I don't know where to start here. Any help would be ...
1
vote
2answers
27 views

A dimension question related to the restriction to a finite-codimensional subspace

Let $V$ be an infinite-dimensional vector space, $T:V \to V$ a linear operator and $W \subset V$ a subspace with $\operatorname{codim} W < \infty$. If $\dim \operatorname{Coker}(T) < \infty$, do ...
0
votes
2answers
31 views

Show that the Open Mapping Theorem requires both spaces to be complete

I am trying to show counter examples to the Open Mapping Theorem. In this particular case, I am trying to show that both spaces need to be Banach. First the OMT: Let $X, Y$ be Banach spaces. Let ...
0
votes
1answer
25 views

Let $X$ and $Y$ be Banach, and $T : X \to Y$ a bounded linear map. When is $T$ closed?

I know that $||T(x)|| \geq C ||x||$ for some $C$ is equivalent. I am looking for less analytic conditions, maybe something to do with the topological structure of $X$. Does anyone know some? By $T$ ...
1
vote
1answer
17 views

Dense and integral zero.

Let $G$ be a compact Lie group and $u\in C^{0}\left(G\right) $. If $\int_{G} u\left( g \right)v \left(g \right)dg= 0$ for every $v\in V $, a subset which is dense in $C^{0}\left(G\right)$, then ...
1
vote
1answer
40 views

Is $C[0,1]$ locally Compact?

I'm asked to use the function $f_n(x)=nx$ for $0\le x\le \frac{1}{n}$ and $f_n(x)=1$ for $\frac{1}{n}\le x\le 1$. I'm not familiar with Functional Analysis.