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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3
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1answer
28 views

Operator norm of an identity map over $l_p$ space

Let $1 \leq p < q \leq \infty$ ($p$ and $q$ are not related) conclude that the identity map I : $ l^n_p → l^n_q$ has operator norm exactly 1. I figured I need to show that given $\|Ix\| \leq ...
0
votes
0answers
26 views

Shift logistic function without moving inflection point from x=0

As a biologist that did not do much maths lately, formulation of my problem may be a bit strange. Sorry in advance and thanks for helping me improving my vocabulary. I am using logistic function from ...
2
votes
1answer
26 views

Norms Equivalence over $\mathbb R^n$

Let $\|\cdot\|$ be any norm on $\mathbb R^n$. Prove that a sequence in $\mathbb R^n$ is Cauchy under the $\|\cdot\|_2$ norm if and only if the sequance is Cauchy under the $\|\cdot\|$. Prove that a ...
5
votes
1answer
31 views

Expanding a norm over a given space

Let $1 \leq p<q \leq \infty$ (p an q are not related) Let $\Phi$ be the vector space of all sequences with at most finitely many nonzero elements, meaning $\Phi=\{\{x_n\}_{n=1}^\infty|$ there is ...
0
votes
1answer
31 views

Is the $l_p$-direct sum of uncountably many separable Banach spaces is separable?

Let $1\leq p<\infty$, let $\Lambda$ be an uncountable index set, and let $(E_\alpha)_{\alpha\in\Lambda}$ be a family of (infinite dimensional) separable Banach spaces. In general, it is known that ...
2
votes
1answer
44 views

projections in von Neuman algebra

Consider a semifinite von Neumann algebra $\mathcal{M}$ with a semifinite faithful normal trace $\tau$. If $Q, P$ are projections in $\mathcal{M}$ with $\tau(Q)< \tau(P)$, then does $\tau(P\wedge ...
7
votes
3answers
126 views

Why are the four fundamental subspaces fundamental?

The four fundamental subspaces in linear algebra, as discussed by Gilbert Strang [1], are the kernel, image, dual space kernel, and dual space image (nullspace, column space, left nullspace, row ...
0
votes
1answer
29 views

Existence of some extension

Let $X_{0}$ be a linear closed proper subspace of real normed space $X$. Show that for every linear and continuous functional $\phi_{0}: X_0 \to \mathbb{R} $ with norm 1 there exist a linear and ...
4
votes
1answer
33 views

To Show Closedness of a Graph in an Application of Closed Graph Theorem

Here's an old exam question I am struggling with: Let E be a Banach space and $ (x_n)_{n \in N} \subset E $ such that $ \sum_{n=1} ^{\infty} | \langle x_n , x^* \rangle | < \infty $ for all ...
0
votes
2answers
16 views

$\bigcap_{S \in L(E,F)} ker(S) = \{0\}$

Let $E$ a Banach space, $F$ a normed space and $L(E,F)$ a set of bounded linear operator from E to $F$. Is true that $$\bigcap_{S \in L(E,F)} ker(S) = \bigcap_{S \in L(E,F)} S^{-1}(0) = \{0\}.$$ If ...
2
votes
2answers
70 views

What do I need to know in advance before taking a course in Functional Analysis?

Do I need a course on measure theory, or could I get by with just picking it up along the way, during the Functional Analysis course? Does the course just use some main results -- lebesgue measure + ...
4
votes
1answer
45 views

Using calculus results for functions of operators

I am interested in the conditions required for functions of operators to be manipulated as if it were a real valued function with a real domain. In an applied maths text I am using the following is ...
0
votes
0answers
32 views

Unit sphere weakly dense in the unit ball

This is an old homework problem from Folland, and I know it has solutions on this website, but I have some questions about the solution provided to us by our TA because there's things about this ...
12
votes
2answers
661 views

Give an example of a real function so that every rational is a strict local minimum

Give an example of $f : \mathbb R → [0, \infty) $ so that every $r \in \mathbb Q$ is a strict local minimum for $f$. Strict local minimum means there is a vicinity $V$ of $r$ such that $f(y) ...
2
votes
1answer
52 views

How can we prove that the space of trace class operators on a Hilbert space $H$ is the closure of $H\otimes H$ with respect to the trace norm?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space over $\mathbb R$ $\mathfrak L^1(H)$ be the space of trace class operators on $H$ and $$\operatorname{tr}L:=\sum_{n\in\mathbb ...
0
votes
2answers
57 views

Show this harmonic function is constant

I'm trying to prove the following let $\alpha \in (0,1)$. If $u \in C^2(\mathbb{R}^n)$ is harmonic and $|u(x)| \leq \|x\|^{\alpha}$, Prove the $u$ is constant. Attempt to prove. Let's observe ...
0
votes
0answers
35 views

Is there a name for these inequalities? Where can I look them up?

Consider the operators $A,B,C$ on Hilbert space $\mathcal H$: Show that: $$ \left \vert \left \vert AB \right \vert \right \vert \le \left \vert \left \vert A\right \vert \right \vert \left \vert ...
2
votes
1answer
41 views

Showing a sequence converges weakly.

Let $f \in L_2(\mathbb{R})$. How can I show that the sequence ${g_n}$ converges weakly to $0$ in $L_2(\mathbb{R})$, where $g_n(x) = f(x − n)$? If this is not true could someone provide a counter ...
1
vote
0answers
16 views

Dual of a point which is in the convex cone of a set, contains the dual cone of that set

Let $\Lambda\subseteq R^n$ contains $m$ elements, where $\lambda_i$ is the $ith$ element, and $co(\Lambda)$ is the smallest convex cone contains $\Lambda$. Also, consider any point $u\in R^n$. Now, I ...
0
votes
1answer
21 views

Functions are in intervals of $L^p$ spaces?

It just occurred to me that a dominated convergence theorem argument proves that $p\mapsto\|f\|_{L^p}$ is continuous, which implies the set of $p$ for which $f\in L^p$ is open. This is shocking to me ...
0
votes
0answers
33 views

What are the Hermitian idempotent matrices with respect to $l_{1}$ norm

Let $A=(M^{n}(\mathbb{C}), \|\cdot\|_{1})$. The numerical range of $a\in A$ is defined as $V(a)=\{f(a):f\in A',\|f\|=1=f(1)\}$. $a\in A$ is said to be Hermitian if $V(a)\subseteq \mathbb{R}$. $a\in A$ ...
0
votes
0answers
36 views

Show that $(\mathbb{R}^n,||\cdot||_2)^*=(\mathbb{R}^n,||\cdot||_2)$

I want to show that $(\mathbb{R}^n,||\cdot||_2)^*=(\mathbb{R}^n,||\cdot||_2)$. So far I know that $$\langle Ax,y\rangle=(Ax)^Ty=x^TA^Ty=x^T(A^Ty)=\langle x,A^Ty\rangle.$$ Could you give a simple ...
1
vote
0answers
19 views

Multiplying the PV$(\frac{1}{x})$ by $x$

I am trying to show that $x\text{PV}\left(\frac{1}{x}\right) = 1$ in the sense of distributions, that is $\langle x\text{PV}\left(\frac{1}{x}\right), \phi \rangle = \langle 1, \phi \rangle$ for all ...
0
votes
0answers
45 views

reference request for $L^p(\partial\Omega)$ in real analysis textbooks

Let $\Omega$ be a bounded open set in $\mathbb{R}^d$. Would anybody come up with a real analysis textbook which contains detailed introductory treatment of the space $L^p(\partial\Omega)$?
4
votes
1answer
41 views

if $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$?

Suppose that $A_n,A$ are self-adjoint operators in $B(H)$. If $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$? From Proposition. 2.3.2 of Pederson'book, I know the result holds in ...
3
votes
0answers
57 views

Existence of Banach Limits

Just want to check everything is good. $\textbf{Theorem:}$ Define $T: l_{\infty}(\mathbb{R}) \to l_{\infty}(\mathbb{R})$ by $$T(x_1,x_2,x_3,...)=(x_2,x_3,x_4,...),$$ $$M=\{x-Tx:x \in ...
1
vote
0answers
41 views

Show that if $P: H\rightarrow H$ is a projection, then $I-P$ is also a projection.

I'm trying to show that if $P: H\rightarrow H$ is a projection, then $I-P$ is also a projection. Would it be enough to show that $$(I-P)(I-P) = I-PI-IP+PP = I-PI-IP+P=I-P-P+P=I-P ?$$ Your help is ...
0
votes
0answers
25 views

Condition for sequential convergence implies convergence in whole.

Let $f:[0,\infty)\to \mathbb R$ be defined as $f(3\cdot2^k)=1$, for $k\in\mathbb N$, and $f=0$ otherwise. Let $a_n=2^n$, for any $s\in [0,\infty)$, $\lim_{n\to\infty}f(a_n+s)=0$, since $\{a_n+s\}$ and ...
0
votes
0answers
40 views

Where does the $L^p$ norm come from?

Where exactly do $L^p$ norms and $L^p$ spaces show up naturally? In other words, how would you arrive at these concepts out of necessity of solving some other problem in a way that motivates using ...
0
votes
0answers
29 views

How to generalize this proof of the closed graph theorem

I found this tricky new proof of the closed graph theorem for a Hilbert space $H$. http://arxiv.org/pdf/1601.02600.pdf It says in the abstract, that it's possible to extend the proof to Banach space. ...
1
vote
1answer
65 views

Trace and norm bounded sequence of positive elements has convergent subsequence in hyperfinite $II_1$ factor

Let $A$ is a hyperfinite $\operatorname{II_1}$ factor and $x_n \in A$ is some sequence of positive elements such that $||x_n||$ convergent and $\operatorname{Tr}(x_n^2) = 1$ (where $\operatorname{Tr}$ ...
1
vote
0answers
33 views

Reference Quest: Measure Theoretic and Functional Analytic Intro to Stochastic Processes

Does anyone have any recommendations for a good book which introduces and cleanly and rigorously explains the measure theory and functional analysis implicit in and relevant to stochastic processes, ...
0
votes
1answer
23 views

How is this a level set? Is it a typo?

I'm reading a proof and it says that if $\phi$ is a continuous linear functional on $L^p[0,1]$ to consider the "level set" $\{f\in L^p[0,1] : \phi(f)\in (-1,1)\}$. I don't get it. Shouldn't a level ...
0
votes
1answer
31 views

Showing that the intersection of two closed linear subspaces is the trivial subspace.

I'd appreciate if someone can provide the best way to deal with this problem. Let $\{\alpha_n\}$ be an orthonormal sequence for a Hilbert space H and let $\{\beta_n\}$ be an orthonormal sequence such ...
1
vote
0answers
29 views

Uniqueness of solution to Helmholtz-style equation

Let $\Omega\subset \mathbb{R}^n$ be open, bounded and connected with $C^1$ boundary. Suppose $q\in L^\infty(\Omega)$ and that $q\geq C>0$ almost everywhere. Consider the following PDE problem: ...
1
vote
1answer
17 views

Convergence in SOT and norm boundedness in $C_r^*(S_\infty)$ equivalent to norm convergence

Let $S_\infty$ - permutation group of the natural numbers fixing all but a finite number of element. And let $C_r^*(S_\infty)$ - reduced group $C^*$-algebra that acts on $\ell_2(S_\infty)$ in obvious ...
1
vote
2answers
48 views

Norm of operator $A$ st. $A^2 = I$?

I'm wondering what can be said about the norm $||A||$ of an operator which squares to identity. All I can think of is that $$1=||AA|| \leq ||A||^2$$ so that $||A|| \geq 1$. But can anything else be ...
0
votes
1answer
22 views

The space $C_0$ does not have a 2-dimensional subspace isometric to $E^2$?

To prove that the space $C_0$ of the sequences x=($x_1,x_2$,...) for which $\displaystyle \lim_{i=\infty} x_i=0$ with the norm $\|x\|=\max_{1 \leq i < \infty} |x_i|$ does not have a two-dimensional ...
-1
votes
0answers
15 views

Find the orthogonal complement of the following space.

let $M=\{ f\in L^2[-1,1] :\int_{-1}^{1} f(t)\ dt = 0\}$ . What is the orthogonal complement of M. Would be really grateful if someone were to point me in the right direction in solving this problem.
4
votes
1answer
47 views

Applying equivalence of norms on $\mathbb R^n$ .

Let $\|\cdot\|$ be any norm on $\mathbb R^n$. Prove that a sequance on $\mathbb R^n$ converges to an element $x \in \mathbb R^n$ under the $\|\cdot\|_2$ norm if and only if the sequance converges to ...
0
votes
0answers
19 views

Does any integral of a function has an inner product form?

It is well-known that the inner product defined on a vector space $V$, is a map: $\langle f, g\rangle : = \int f(x)g(x) dx$, for all $f,g \in V$. My question is if any integral has an inner product ...
2
votes
2answers
39 views

Schauder bases of subspaces of the sequence space $\ell^p(\mathbb{N})$

Consider the canonical Schauder basis $\{e_i:i\in \mathbb{N}\}$ for $\ell^p(\mathbb{N})$, where $e_i(j)=\delta_{ij}$. Let $M$ be a subspace of $\ell^p(\mathbb{N})$. Is it right that $\{e_i:i\in ...
2
votes
1answer
21 views

a basic question in crossed product for compact group action

I am quite new into crosssed product of Fréchet algebras or C$^*$-algebras. So if the question is too basic please excuse me. Suppose we have two Fréchet algebras or C$^*$-algebras $A$ and $B$ and ...
0
votes
1answer
48 views

$\ell_2$ convergence and $\ell_1$ norm convergence implies $\ell_1$ convergence

Let $x_n \in \ell_2$ converge to $x_\infty \in \ell_2$ and $||x_n||_1$ converge to $||x_\infty||_1$ where $||\cdot||_1$ is $\ell_1$ norm. Is it true, that $x_n$ converge to $x_\infty$ in $\ell_1$?
0
votes
0answers
12 views

Find conditions on operator $A$ , {$Af_n$} is in $L^2$ and $\lim\limits_{n\mapsto \infty} \int_a^b (Af_n - f)^2 dx =0$.

Consider a linear operator $A$ . Please could you state sufficient conditions on $A$ other than the one I gave such that for any $f$ in $L^2$ there is a sequence {$f_n$} such that the sequence ...
0
votes
0answers
23 views

A estimate about constant coefficient partial differential operator on $C_0^{\infty}(\Omega)$

This problem is from Stein Real Analysis,Chapter 5,exercise 12. Problem: We consider whether the inequality $||u||_{L^2(\Omega)} \le c||Lu||_{L^2(\Omega)}$ can hold for open sets $\Omega$ that are ...
2
votes
1answer
28 views

Unique ground state of Schrödinger Operators

I'm reading a book and there is an argument that the ground state of a Schrödinger operator is unique. The problem is I think the argument is complete non-sense! These are lecture notes by Witten, I ...
2
votes
1answer
36 views

spectral projection of an isolated point in the spectrum of a closed linear operator

Suppose that $T$ is a closed densely defined operator on a Hilbert space $H$ with $\rho(T) \neq \emptyset$. If $\lambda \in \sigma(T)$ is an isolated point then we know that $H = \mathcal{N}(E_o) ...
2
votes
1answer
36 views

Weak convergence and strong convergence on $B(H)$

Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence ...
3
votes
1answer
17 views

Positive maps and $*$-homomorphisms

If $\varphi:A \to B$ is a linear map between $C^*$-algebras, it is said to be positive if it sends positive elements in $A$ to positive elements in $B$. We know that every $*$-homomorphism is ...