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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385 views

Direct sum of a closed linear subspace of a Hilbert space

I am currently reading a proof that involves some result from functional analysis which I would like to understand a little better - Suppose we have a Hilbert space $\mathcal{H}$ and a closed linear ...
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1answer
1k views

Proving that closed (and open) balls are convex

Let $X$ be a normed linear space, $x\in X$ and $r>0$. Define the open and closed ball centered at $x$ as $$ B(x, r) = \{y \in X : \Vert x − y\Vert < r\} $$ $$ \overline{B}(x, r) = \{y \in X : ...
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1answer
117 views

Limit inferior taken on the norm of a sequence

Let $E$ a normed vector space and let $(x_n)$ be a sequence in $E$. Suppose that $x_n$ converges weakly (i.e. wrt the weak topology) to $x$. Why is it that from the inequality $$ |f(x_n)| \leq \|f\| ...
7
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1answer
741 views

What is the predual of $L^1$

Is there a nice characterization of the predual of $L^1$? So, what does the space $X$ look like, such that $X^*=L^1$, where the star denotes the dual of a space. How do you start to find such preduals ...
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1answer
1k views

Direct aproach to the Closed Graph Theorem

In the context of Banach spaces, the Closed Graph Theorem and the Open Mapping Theorem are equivalent. It seems that usually one proves the Open Mapping Theorem using the Baire Category Theorem, and ...
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67 views

Making a space complete

If I have a homeomorphism between $X$ and $Y$. Suppose $X$ is complete and $Y$ is incomplete under the same metric, how can I make $Y$ complete using the fact that it is homeomorphic to $X$. Consider ...
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1answer
316 views

The exponent of self-adjoint operator

If $X$ is a Hilbert space and $A$ is an unbounded self-adjoint operator on $X$, is it necessarily that $A^2$ is self-adjoint as well?(admittedly, $A^2$ is densely defined)
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1answer
70 views

Domain and range of the function in Devaney's definition of chaos

Devaney's definition of chaos assumes the function $f: V \rightarrow V$. Is there a reason for restricting this to $V \rightarrow V $ and not two different sets $X\rightarrow Y$?. Specifically, my ...
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227 views

Why is the numerical range of a self-adjoint operator an interval?

I was reviewing for a test for functional analysis when I came across the following statement: Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. Then the numerical range of it is ...
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3answers
393 views

Understanding Arzelà–Ascoli theorem

I am going through AA theorem and its proof as per wikipedia : http://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem, I am not able to understand clearly the formation of diagonal ...
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1answer
409 views

Is this operator compact?

Suppose ($x_n$) is a normalized, linearly independent, sequence in a reflexive Banach space $X$, and $T$ is an injective, strictly singular, bounded operator on $X$ such that $Tx_n\longrightarrow ...
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1answer
82 views

comparison between spaces

There a lot of function spaces and would be nice if somebody can correct me if I am wrong in comparing a few. I want to compare $C^2,L^2,W^{2,2}$ (continuous up to third derivative, Hilbert space of ...
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0answers
217 views

Why is this functional coercive?

Let $h\in L^2(0,1)$, such that $|\int h \mathrm{d}x|<1$. On the space $H^1(0,1)$, consider $$J(v)=\frac{1}{2}\int_0^1 v'^2\,\mathrm{d}x+\int_0^1\sqrt{1+v^2}\,\mathrm{d}x-\int_0^1hv \,\mathrm{d}x.$$ ...
3
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1answer
230 views

Show $L^2$ is not closed in $L^1$

Suppose $X \subset L^1_{([0,1])}$ is the subspace consisting of all square-integrable functions. I have to show that $X$ is not a closed subset of $L^1_{([0,1])}$. How do I go about doing this? What ...
3
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1answer
252 views

Compact operator in Hilbert Space

$H$ is a Hilbert space and $A$ is a bounded operator on $H$. If $A^*A$ is compact, is it necessarily that $A$ is compact?
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1answer
365 views

Weak convergence in $L^2$ and uniform covergence

I have this problem: let $f_n$ converge weakly to $f$ in $L^2[0,1]$ and let $$F_n(x)=\int_0^xf_n(t) \, \textrm{d}t,$$ $$F(x)=\int_0^xf(t) \, \textrm{d}t.$$ Then $F_n,F$ are continuous and $F_n$ ...
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1answer
168 views

how to show $f$ attains a minimum?

Let $H$ be a Hilbert space and let $f\colon H\rightarrow \mathbb{R}$ be a continuous convex function such that $f(x_n)\rightarrow\infty$ whenever $\lVert x_n\rVert\to\infty$. We need to show that $f$ ...
2
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0answers
143 views

Frames and completeness

Let $H$ be a separable Hilbert space. It is known that if $\{f_{n}\}$ is a frame then it is complete, but the converse is not true. In which cases the converse will be true, i.e. When a complete ...
2
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1answer
830 views

weak derivative of a nondifferentiable function

I am reading a book on Sobolev and having trouble understanding a notion of weak derivative. I consider a function $(x-1)^+=\max(x-1,0),x\in[0,2]$, I have a problem at $x=1$, so it is continuous and ...
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2answers
2k views

The kernel of a continuous linear operator is a closed subspace?

If $V$ and $W$ are topological vector spaces (and $W$ is finite-dimensional) then a linear operator $L\colon V\to W$ is continuous if and only if the kernel of $L$ is a closed subspace of $V$. ...
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2answers
394 views

Approximate eigenvalue and continuous spectrum

Let $\mathcal{H}$ be a Hilbert space and let $A: \mathcal{H} \rightarrow \mathcal{H}$ be a bounded operator. While studying different definitions of the continuous spectrum of $A$ (one using ...
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1answer
123 views

Quotient space and $L_p$ space

Let $Tx(t)=x'(t)$ in $L_{p}(0,1)$, $1\leq p<\infty$. $\textrm{Dom}(T)=\left\{ x\in L_{p}(0,1):x\in AC[0,1],x'\in L_{p}(0,1)\right\}$. Since the Lebesgue measures of $\{0\}$ and $\{1\}$ are zero, ...
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The direct sum of two closed subspace is closed? (Hilbert space)

I know that if $X$ is a Banach space, then, the direct sum of two closed subspace $X_1$ and $x_2$ is not necessarily closed. But what if $X$ is Hilbert? I assume there is something to do with the ...
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1answer
610 views

Compact multiplication operators

In class, we started talking about operators on Banach spaces after covering the Arzela-Ascoli Theorem. We defined a continuous operator $T\colon X \to Y$ to be compact if $\overline{T(B_X)}^{Y}$ is ...
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1answer
142 views

How to derive this inequality from Peetre's Inequality

Suppose we are given the following inequality, which holds for any real number $s$ and $x,y \in \mathbb{R}^n$: \begin{equation} (1 + |x + y|)^s \leq (1 + |x|)^s(1 + |y|)^{|s|} \end{equation} (This ...
3
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1answer
187 views

Inner product for vector - valued functions

I understand that, for example the inner product space $L^2(X)$ of complex - valued functions defined on $X$ has the inner product \begin{equation} (f,g) = \int f \, \overline{g\,}. \end{equation} ...
2
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1answer
128 views

$f_n\rightarrow f$ in $L^1$ implies $gf_n \rightarrow gf$ in $L^1$ for smooth $g$?

Suppose I have some $f\in L^1$, such that $x^k f(x)$ is also integrable. Now I have some $\{f_n\}\subset\mathcal{C}^\infty$ satisfying $f_n\rightarrow f$ in $\lVert\cdot\rVert_1$. Is it true that $x^k ...
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2answers
110 views

Compactness in $L^1$

Let $\mu(\cdot)$ be a probability measure in $X$. Consider a function $f: Z \times X \rightarrow \mathbb{R}_{\geq 0}$, with $Z \subset \mathbb{R}^n$ compact, such that: $\forall x \in X, \quad z ...
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1answer
55 views

Finding a vector in a n.l.s.

Let $X$ be a normed linear space and $Y$ a closed proper subspace. Prove that for all $\varepsilon > 0$, there is an $x \in X$ with $\|x\| = 1$ and such that $\|x − y\| ≥ 1 − \varepsilon$ for all ...
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2answers
159 views

Is it true that $\|f \|_{L^{p-1} }\leq \|f\|_{L^{p}}$?

Is this true? $$ \|f \|_{L^{p-1} }\leq \|f\|_{L^{p}}\;\; $$ Specifically I know $\;\;\|f\|_{L^{2}} \leq \|f\|_{L^{\infty}}$ $\;$ but I can't figure out why?
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2answers
118 views

Why is the canonical injection always continuous from $\sigma(E,E^*)$ into $\sigma(E^{**},E^*)$?

I am told that this is because for a fixed $f \in E^*$, $x \mapsto \langle Jx, f \rangle = \langle f, x \rangle$ is continuous wrt $\sigma(E,E^*)$. Now, I understand this is the case. But why does ...
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1answer
166 views

Homogeneous and rotational invariant distribution

If $u \in \mathcal D'(\mathbb R^n)$, $u$ is homogeneous of degree $0$ and rotational invariant, it is necessarily that $u$ is a constant? (Since if $u \in C^\infty$, the conclusion obviously hold.)
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1answer
953 views

Must a complete space be locally compact?

There are two versions of second category space: one is complete metric space, the other is locally compact space. As we know, an open interval is locally compact but not complete. But how about the ...
3
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1answer
308 views

Prove that the set of extreme points in $B$ is equal to an atom

How to prove this? Please help me. Thank you very much. A measurable set $E$ in a measure space $(X, \mathcal{M}, \mu)$ is said to be an atom if $\mu (E) > 0$ and no proper measurable subset of ...
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1answer
208 views

Example of strictly convex space with not strictly smooth dual

I'm trying to find an example of a space $V$ which is strictly convex, but has a dual space $V^*$ which is not strictly smooth. Any help please?
5
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1answer
794 views

Why is the dual space of $H_0^1(\Omega)$ denoted $H^{-1}(\Omega)$?

Why is the dual of the Sobolev space $H_0^1(\Omega)$ denoted $H^{-1}(\Omega)$ ? For a positive integer $k$, $H^k(\Omega)=W^{k,2}(\Omega)$. What is the motivation behind the $-1$ exponent?
2
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1answer
80 views

Finding an ON basis of $L_2$

The set $\{f_n : n \in \mathbb{Z}\}$ with $f_n(x) = e^{2πinx}$ forms an orthonormal basis of the complex space $L_2([0,1])$. I understand why its ON but not why its a basis?
5
votes
1answer
136 views

Prove that if f in $C(X \times Y)$ then there exists functions.

Let $X$ and $Y$ be compact metric spaces. I am trying to prove that if $f \in C(X \times Y)$ and $\varepsilon > 0 $, then there exist functions $g_1, g_2,...,g_n \in C(X)$ and $h_1,...h_n \in C(Y)$ ...
2
votes
1answer
201 views

Norm closure of convex hull of its set of extreme points

How to prove that the set of extreme points of $B_{\ell^1} = \{v \in \ell^1 : \| v \| \le 1\}$ is $\{ +e^N, -e^N : N=1,2,3,\ldots \}$, where $e^N$ denotes the Nth standard basis element in $\ell_1$: ...
3
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1answer
316 views

Is there a solution to this integral equation?

The problem is related to this question: How to find eigenfunctions of a linear operator (follow-up question) I posted earlier. Suppose I want to solve the following integral equation: $$\int_0^1 ...
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2answers
319 views

The spectrum of a bounded linear operator

Suppose $X$ is a Banach space. For $T\in L(X,X)$, let its spectrum be $\sigma(T)$. Show that $\lambda\in\sigma(T)\Rightarrow\lambda^{n}\in\sigma(T^{n}),\ \forall n\in\mathbb{N}$. Show that the ...
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1answer
766 views

Bounded inverse operator

Let $X$ and $Y$ be Banach spaces. Suppose $T$ is a linear operator from $X$ onto $Y$ with $\operatorname{Dom}(T)\subset X$. Show that $\exists T^{-1}\in L(Y,X)\Leftrightarrow\exists M>0:\ ...
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2answers
492 views

Convex hull of the extreme points of the unit ball in $C(K)$ where $K$ is the Cantor set.

Let $K$ denote the usual $1/3$ Cantor set and let $B=B_{C(K)}$ (here $B_{C(K)}$ = {$v \in C(K) : \|v\| \leq 1 $} denotes the closed unit ball of $C(K))$. Then how to prove that $B$ coincides with the ...
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1answer
132 views

Stone-Cech compactification

Please help me to figure out this problems. Let X be a metric space and let Y be a compact metric space. Denote by $\Phi: X \rightarrow \beta X$ the standard embedding of X as a dense subset of its ...
2
votes
2answers
693 views

Showing that $\ker T$ is closed if and only if $T$ is continuous. [duplicate]

Possible Duplicate: $T$ is continuous if and only if $\ker T$ is closed Let $T: X\to \mathbf{R}$ be linear. Suppose that $X$ is a Banach space. I want to show that $T$ is continuous if ...
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1answer
515 views

Is $[0,1]^\omega$ homeomorphic to $D^\omega$?

Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case. This observation leads to ...
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0answers
195 views

Elliptic PDE regularity

I have $u \in H^1_0(\Omega)$ where $\Omega$ is bounded which solves some elliptic PDE of the form: $$-\Delta u + h(u) = f$$ in $\Omega$, where $f \in L^2(\Omega)$ $$u = 0$$ on $\partial\Omega$, say. ...
2
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2answers
581 views

Why is Parseval's Equality and Bessel's Inequality Different?

Bessel's Inequality: $\sum_n |\langle x, e_n \rangle |^2 \leq \|x\|^2$ Parseval: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\sum_n |\langle x, e_n \rangle |^2 = \|x\|^2$
2
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1answer
465 views

Yet another exercise from Stein's Real Analysis

So I'm stuck at the following result, about compact operators on Hilbert spaces (which I think it's called Fredholm's theorem) from Stein. It's exercise 29 from Chapter 4. Let $T$ be a compact ...
0
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1answer
105 views

Help with understanding a proof from Brezis

I am reading Brezis's book and I have trouble understanding a proof. I put up an image of the relevant part of the text. What I don't understand is towards the end when he writes: "It follows from ...