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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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 ...
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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. ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 $\bar{a}(v,w)$ ...
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2answers
42 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 ...
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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 ...
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1answer
26 views

Approximate eigenvalues of an ergodic invertible transformation

Consider a non-atomic probability space $(X,\mathcal{B}, m)$. Let $T: X \to X $ be an ergodic invertible measure preserving transformation.Let $U_T$ be the Koopman operator associated with $T$. Show ...
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1answer
17 views

There is a relationship between the norm of a linear operator T and the values it takes on a space base?

Let $T$ be a bounded linear operator on a infinite dimensional Hilbert space $H$ and let $B$ be an bases of $H$ Is there a relationship between the norm of $T$ and values in $T(B)$?
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1answer
40 views

Norms inequality in a sequence 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 ...
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1answer
30 views

Baire class one function in terms $\varepsilon-\delta$

in here @Brian M. Scott has proved that a function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\sum_{q_n\leq x} \frac{1}{2^n}$, where $\mathbb{Q} =\{q_1, q_2, \cdots\}$, is Baire one. In this paper ...
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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 ...
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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
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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
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1answer
34 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 ...
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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 ...
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2answers
71 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
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1answer
46 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 ...
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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 ...
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2answers
665 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) ...
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1answer
53 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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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 $V(a)\subseteq \mathbb{R}$. $a\in A$ ...
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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 ...
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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 ...
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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 $L^p(\partial\Omega)$?
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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 ...
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59 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 ...
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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 ...
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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 ...
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41 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 ...
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30 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. ...
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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 $\operatorname{Tr}$ ...
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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, ...
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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 ...
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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 ...
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0answers
30 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: ...
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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 ...