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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operator in a hilbert space

can someone plese help me to answer the following problem: Let (ek) be a total orthonormal sequence in a separable Hilbert space H and Let T: H→H be defined at ek by: T(ek) = ek+1. , k = 1, ...
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2answers
32 views

Prove $\ell^5$ is contained in $\ell^6$.

I am struggling with the proof to show that, for any $p$, $r$ such that $1 \le p <r < \infty$, that $\ell^ p\subset\ell ^r$. Could somebody please give a helpful nudge by showing how this ...
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1answer
28 views

Prob. 3, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal? A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x ...
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1answer
27 views

Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
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1answer
22 views

Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
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0answers
12 views

Calculating the norm of a case dependent function

If $X$ is the Hilbert space $L^2(0,\infty)$ equipped with the inner product $$\langle f,g\rangle :=\int_0^\infty f(\zeta)\overline{g(\zeta)}(e^{-\zeta}+1) \, d\zeta,$$ and the operator $T(t):X\to ...
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5 views

Generating a contraction semigroup on an energy space

Consider the system of partial differential equations $\displaystyle\frac{\partial Q}{\partial t}(\zeta, t)=-\frac{\partial}{\partial\zeta}\frac{\phi(\zeta, t)}{L(\zeta)}$ ...
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0answers
16 views

Fourier transform and series

Let $f \in L^2(\mathbb{R})$ and $F(f|_{[m,m+1]})$ be the Fourier transform of a restriction of $f$. Does this imply that $$\sum_{m,n \in \mathbb{Z}} |F(f|_{[m,m+1]})(2 \pi n)|^2 $$ exists and is ...
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0answers
15 views

Schrodinger Operator with Finite Discrete Spectrum in $(-\infty, -1]$

I'm reading parts of Reed and Simon's Analysis of Operators and have come across a statement I find puzzling. They say that if $V$ is a bounded function of compact support on $\mathbb{R}^3$ then ...
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1answer
7 views

Supporting hyperplane of convex function

Below is the appendix B of Evan's PDE book on supporting hyperplanes of convex functions. In the remark (1), he says that the mapping $y\to f(x)+r\cdot(y-x)$ determines the supporting hyperplane to ...
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13 views

An exersice about Isomorphic Hilbert Spaces and the Fourier Transform for the Circle [on hold]

An exersice from section 5 of conway's functional analysis
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0answers
10 views

When do closed subspaces of a Banach space fit together nicely?

Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace ...
2
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1answer
23 views

closed graph theory and unbounded operator

I am studying unbounded operators and the graphs of those operators. I found that the closure of a graph may not be the graph of any operator. Can someone provide an example of an operator and a ...
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1answer
33 views

The proof of finding extreme points of the unit ball of $l^1$

Can someone show how to start the proof of finding extreme points of the unit ball of $l^1$? Thanks. Edit: How I've done so far is that Let $B$ be the closed unit ball of $l^1$ Consider any $x_n ...
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1answer
11 views

properties of orthonormal systems and hilbert spaces [on hold]

I need to show (a) $\implies$ (b) For an orthonormal system $\{\phi_i\}_{i=1}^\infty$, and a Hilbert space $H$, the following are equivalent: (a) If $\langle f,\phi_i\rangle=0$ $\forall i$, ...
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1answer
15 views

Polarization Identity: Sesquilinearity

Problem Given a vector space $V$. Consider quadratic forms with: $$q[u+v]+q[u-v]=2q[u]+2q[v]$$ Then one has a 1-1-correspondence: $$q_s[v]:=s(v,v)\quad ...
2
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2answers
61 views

What is the largest function whose integral still converges?

Let C be the set of all functions $f(x)$ whose integral converges, i.e. for some constant $x_0$: $$\int_{x_0}^\infty f(x) dx < \infty$$ While playing with integrals in Wolfram Alpha, I noticed ...
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0answers
13 views

fractional powers in Banach algebra [on hold]

Let $X$ be a Banach algebra. For $x\in X$ and $0< p< 1$, would $x^p\in X$? If not, under what conditions it holds?
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0answers
18 views

Extreme points of the unit balls of $l^\infty, C([0,1])$

Determine the extreme points of the unit balls of $l^\infty$, and $C([0,1])$ for real-valued functions, with the uniform norm. Is $C([0,1])$ the dual of a Banach space? I've found the extreme points ...
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1answer
17 views

Diffeomorphism preserves compact support of functions?

Let $M$ and $N$ be two Riemannian manifolds which are diffeomorphic via a $C^k$ map $F:M \to N$. Let $\phi \in C^0_c(M)$ be a continuous function with compact support in $M$. Is it true that its ...
5
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3answers
76 views

The function $\phi(p)=\|f\|_{L^p}^p$ is convex

Fix an arbitrary function $f\in L^p([0,1])$ and define $$\phi(p)=\|f\|_{L^p}^p$$ for $p\in [1,\infty)$. Prove $\phi$ is convex. Comments: This is a standard property of $L^p$ spaces, but no ...
2
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2answers
56 views

The $L^p(\mathbb R)$ norm is increasing as a function of $p$ (Update: It's false!)

Update: This is false. See the answers for a counterexample. Let $C\ge 1$ be a constant. Fix $f\in L^p(\mathbb R)$ for $p\ge C$. Show that $$p\rightarrow \left( \int |f|^p \right)^{1/p}$$ is ...
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0answers
31 views

Under what condition does $f$ belong to $L^p\left(\mathbb{R}^n\right)$? [on hold]

Let $a,\,b>0$ and $f(x) =(1+|x|^a)^{-1} + (1+|\log|x||^b)^{-1}$ Under what condition does $f$ belong to $L^p\left(\mathbb{R}^n\right)$?
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0answers
38 views

Common traits of functions which are non-trivial to integrate?

My question is very simple: do there exist certain qualities of functions such that functions which possess these qualities are guaranteed not to have anti-derivatives which are expressable in terms ...
4
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0answers
39 views

Is $C([0,1])$ for $\mathbb{C}$ dual to any Banach Space?

I've been able to show that the extreme points of $C([0,1])$ are the continuous functions that take values on the unit circle. However, I'm not sure how to reason from here as to whether or not it is ...
3
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1answer
30 views

Intuition behind the Riesz-Thorin Interpolation Theorem

Quoting the definition on Wikipedia, Let $(\Omega_1, \Sigma_1, \mu_1)$ and $(\Omega_2, \Sigma_2, \mu_2)$ be $\sigma$-finite measure spaces. Suppose $1 \leq p_0 \leq p_1 \leq \infty$, $1 \leq q_0 ...
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0answers
30 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
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22 views

a question about prove an exponential matrix function can be infinitely differentiable

If I have an exponential matrix exp(t(U+sH)), can someone tell me what is the dirivative with respect to s? I am really confused. (where U and H are matrices,and s,t are real numbers). Thus,if I let ...
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2answers
29 views

Extreme points of unit ball of Banach spaces $\ell_1$, $c_0$, $\ell_\infty$

Find extreme points of the unit balls of each Banach space, $l^1 $, $c_0$, $ l^\infty$ Can you help me with this one? For the first space, $l^1$, I thought there was no extreme point, but ...
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1answer
39 views

Why using liminf instead of limsup?

In Chapter 8: Calculus of variations of Evan's Partial Differential Equations, Evan writes as follows: I am wondering about the last paragraph where he says that knowing $I[u] \leq ...
1
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1answer
15 views

Numerical range of inverse operator

Let $T$ be a bounded self-adjoint operator such that the numerical range is contained in $[a,b]$ with $0<a<b< \infty.$ Does it then follow that the numerical range of $T^{-1}$ is contained in ...
2
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1answer
22 views

Continuity of functionals on Sobolev space

Let $U$ be a bounded set in $R^n$ and $W^{1,p}(U)$ denote a Sobolev space. Suppose $\{w_n\}\subset W^{1,p}(U)$ converges to $w \in W^{1,p}(U)$. Let $I[w]=\int_U F(Dw,w,x)dx$ for $w\in W^{1,p}(U)$, ...
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1answer
26 views

Closure of bounded set is bounded? Topological space

Let $X$ be topological space, and $A \subseteq X$ that is bounded. Is the closure of $A$ also bounded? This is true if $X$ is topological vector space, but is it if $X$ is only topological?
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Measureable functions and its properties [on hold]

I have two question about measureable functions and its properties and I want some help to solve them $1)$ if $f$ and $g$ are positive measureable functions then $f-g$ is measureable function ? ...
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0answers
18 views

Using Bounded Operator sequence Theorem

Let E$\subset L_1$ be a set of fourier series functions $e_n(t)=e^{int}$ for $n \in Z$. What is meant by saying to prove $Ge_n$ is a scalar multiple of $e_n$ and it is continuous? How can we prove it? ...
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0answers
19 views

DFT and circular convolution

Given sequence $x[n]=\{1,1,1,1,3,3,3,3,1,1,1,2,2,2,2,1,1,1,1\}$ and a filter $h[n]=\{1,-1,1,-1\}.$ These are circularly convolved using $19$ point Discrete Fourier transform (DFT). What is the out ...
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1answer
42 views

Why every algebra on finite set is a topology

How can I prove that every algebra on finite set is a topology on this set And if the set is infinite how can give me an example algebra but it isn't topology
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1answer
15 views

How can I prove that in monotone class

How can I prove that : Let $X$ be nonempty set and $A$ is algebra in $X$ and $A$ is a monotone class , then $A$ is $σ$ Algebra in $X$
2
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0answers
32 views

How can I solve like this exercise about measurable function

How can I solve the following exercise Every positive measurable function is limit of increasing sequence of positive simple function How can I prove that I need the proof with explain or how can ...
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1answer
29 views

can a LUB be part of an open interval

I am re-reading an old textbook "Introduction to Hilbert spaces and applications" by Lokenath Debnath and Piotr Mikusinski, and there is a proof of a lemma in a chapter about the Lebesgue integral ...
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0answers
24 views

Weak convergence of a sequence of elements of a compact set implies strong convergence

Let E be a Banach space and $K \subset E$ a compact subset in the strong topology. Let $(x_n)_{n \geq 1} \subset K$ such that $x_n \rightharpoonup x$ in $\sigma (E, E^*)$. K is compact, so $(x_n)_n$ ...
3
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0answers
33 views

Prove that range of operator is closed.

$X$ and $Y$ are Banach spaces and $T$ is a bounded linear operator from $X$ to $Y$ which sends bounded closed sets to closed sets. Prove that $T(X)$ is closed. Here I tried to used the fact that ...
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1answer
37 views

Hahn-Banach Thm for Normed Space.

Let $X$ be a normed Space. For $x \in X$ define $J(x)=\{f \in X^ * : f(x)=\|x\|^2 , \|f\|=\|x\|\}$. Prove that $J(x)$ is not the empty set.
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1answer
20 views

Parseval's identity does not hold for constucted basis

As part of an exercise, I was asked to show that given an orthonoraml basis $(\varphi_1,\varphi_2,\varphi_3,...)$ in $L_2[-\pi,\pi]$, we can construct an orthonormal basis $(\psi_1,\psi_2,\psi_3,...)$ ...
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1answer
15 views

Equivalent definition of bounded set in norm linear space

Definition of Bounded set in norm linear space. If $X$ is norm linear space and $B \subseteq X$, then $B$ is bounded if there exists $M>0$ such that $\sup_{n\in \mathbb N} \lvert \lvert x_n ...
2
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1answer
22 views

Adjoint operator on Banach space

Suppose $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded linear operator. Show that $T$ is an isometric isomorphism if and only if its adjoint $T^*$ is also an isometric isomorphism. Given an ...
2
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1answer
57 views

All derivations are directional derivatives [duplicate]

Let $X : C^{\infty}(\mathbb{R}^n) \rightarrow \mathbb{R}$ be a derivation, so i.e. linear and satisfying the Leibniz Rule $$X(fg)=X(f) \cdot g(a)+X(g) \cdot f(a)$$ for some fixed $a \in ...
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0answers
16 views

unbounded operator and closed graph theory [on hold]

on this ex we want to prove that the closed graph is not always an graph for linear operator how can i solve this problems any hints this problem is taken from reed-Simon book how can i prove ...
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1answer
24 views

Why is this inequality true?

In Evan's Partial Differential Equations, he writes Then, he continues to write: But I do not understand how he gets $I[w] \geq \delta ||Dw||^q_{L^q(U)} - \gamma$. I tried to write it out and I ...
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0answers
9 views

Normal Operators: Backtransform

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$W:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{W}$$ By a ...