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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Question about inequality relating infinity norm and Lebesgue integral

I have a question about the following question: Let $f:\mathbb{R} \to \mathbb{R}$ be a measurable function. We know there exists a constant $K$ such that for every bounded continuous function ...
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35 views

Consider the space $L^2$ and let $V(x)(t)=\int\limits_0^tx(s)ds$ be a linear operator, what is the conjugate operator $V^*$ of $V$?

Consider the space $L^2[0,1]$ and let $V(x)(t)=\int\limits_0^tx(s)ds$ be a linear operator from $L^2$ to $L^2$, what is the conjugate operator of $V$? I know the dual space is the same, and if $g\in ...
4
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63 views

Ranges of projection operators

Suppose that $X$ is a Banach space and $P$ and $Q$ be bounded linear projections on $X$ such that $PQ$ and $QP$ are compact. Does it follow that $PQ$ and $QP$ are finite-rank operators? My attempt: I ...
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17 views

$X,Y$ Banach spaces, $A\in B(X,Y)$, $\text{ran}A$ second category, then $\text{A}$ is closed. [duplicate]

this is question $III.13.1$ from Conways book on functional analysis: Suppose $X,Y$ and Banach spaces, $A\in B(X,Y)$, $\text{ran}A$ is a second category space, then $\text{ran}A$ is closed. ...
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32 views

Prove or disprove that this space is complete

Consider the function $||\cdot||_\infty:l^2\to[0,\infty)$ given by $$||(a_n)||_\infty = \sup\{|a_n|:n\ge 1\}.$$ so I want to figure out if $ (l^2,||\cdot||_\infty)$ is complete. Then I was wondering ...
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23 views

If $f_n \rightharpoonup f$ and $g_n \rightharpoonup g$, and $|f_n|_H - |g_n|_H \to |f|_H-|g|_H$, does $f_n \to f$ and $g_n \to g$?

We work in a Hilbert space $H$. If $f_n \rightharpoonup f$ and $g_n \rightharpoonup g$, and $$|f_n|_H - |g_n|_H \to |f|_H-|g|_H$$ is there any chance that $f_n \to f$ and $g_n \to g$? Of course this ...
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1answer
55 views

$L^p$ and $L^1$ norms

As $L^p$ norm of $f:\mathbb R\to \mathbb R^{n}$ is defined as $$ \|f(t)\|_p:=\left(\int_S\|f(t)\|^pdt\right)^{1/p} $$ I have two questions: What kind of norms for the function $f(t)$ in the ...
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39 views

Why the measure of the Cameron-Martin space is zero?

I am studying the construction of an abstract Wiener space on "Kuo - Gaussian measures in Banach spaces". Consider an abstract Wiener space $(X, H, \mu)$ where $X$ is a real separable Banach space, ...
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1answer
50 views

Application of Hahn Banach Separation theorem

I am solving an exercise (not Homework).. Let $E_1$ and $E_2$ be non empty disjoint convex subsets of $X$, with $E_1$ compact and $E_2$ closed in $X$. Then there are $f\in X'$ and $t_1,t_2$ in ...
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2answers
17 views

Is this a property of Normed Vector Spaces?

Let $X$ a normed vector space on $(\mathbb{K}, + , . )$. Is the following assertion true? Any $x$ of $X$ can be written as $x = \alpha a$ , $\alpha \in \mathbb{K}$ , $a \in X$ with $||a||_X=1$ ...
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48 views

Prove a map of binary expansion is continuous

Prove that the map: f: $\{0,1\}^\mathbb{N} \times \{0,1\}^\mathbb{N}$ $\to$ $[0,1] \times [0,1]$ is continuous. I know that the map can be written as ($m_1,m_2,m_3,m_4,...$) $\times$ ...
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1answer
23 views

Let $f$ and $g$ linear operators where $f$ and $g$ commute and $f$ has simple spectrumm, then there is $P$ a polynomial such thah $g=P(f)$.

Let $f : \mathbb{C}^{n}\rightarrow \mathbb{C}^{n}$ be a linear operator with a simple spectrum, furthermore, let $g : \mathbb{C}^{n}\rightarrow \mathbb{C}^n $ be a linear operator such that $f$ and ...
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1answer
66 views

When does weak convergence of a sequence imply almost everywhere convergence of Cesaro averages?

Fix $(X,\mu)$ a nonatomic probability space, $U\colon L^2(X,\mu) \to L^2(X,\mu)$ a unitary operator and $(r_n)$ some fixed sequence such that $U^{r_n}\to V$ in the weak operator topology: $\langle ...
4
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65 views

Trace-class, Hilbert Schmidt operators, $L^p(H)$: duality theorems

Let $H$ be a Hilbert space, separable if necessary, and let $tr$ be the usual trace on $L^1(H)$. It is classical theory that $K(H)^*=L^1(H)$, and $L^1(H)^*=B(H)$, via the canonical application ...
6
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28 views

Banach algebra, map $f \mapsto {1\over{2\pi i}} \int_S f(z) \cdot (z - a)^{-1}dz$ well-defined?

Let $A$ be a Banach algebra over $\mathbb{C}$ and $N: A \to \mathbb{R}_{\ge0}$ the corresponding multiplicative norm. For any $a \in A$, we define$$\text{spec}(a) = \{\lambda \in \mathbb{C} : \lambda ...
7
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54 views

For any ideal $I \subsetneq \mathbb{C}(X)$ the set $\{x \in X: f(x) = 0 \text{ for all }f \in I\}$ is not empty?

See here for a related question by someone else. Let $X$ be a compact metric space and $\mathbb{C}(X)$ the algebra of continuous functions $f: X \to \mathbb{C}$, with pointwise operations. Does it ...
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37 views

Example (for some $X$) of a nonclosed ideal in $\mathbb{C}(X)$?

Let $X$ be a compact metric space and $\mathbb{C}(X)$ the algebra of continuous functions $f: X \to \mathbb{C}$, with pointwise operations. We equip $\mathbb{C}(X)$ with the maximum norm $N(f) := ...
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30 views

How to generate weak topology by the family of semi-norm?

How to generate a weak topology by the family of semi-norm $$ \{P_i \colon A \rightarrow B, i \in I\} $$ in which $$ P_i(a)=\lVert ia\rVert+\lVert ai\rVert$$ when $A$ is a Banach algebra and $I$ is ...
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2answers
32 views

convergence in $L^p, p >2$ and convergence in $L^1$

Let $f_n$ be a sequence of functions in $L^p, p>2$, and let $f$ be also in $L^p$. In order to show $f_n \to f$ in $L^p$, I would need to show $\int |f_n - f|^p \to 0 $. Instead of showing that, ...
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1answer
136 views

domain of heat semigroup,

In Example 7.20 p.99 of enter link description here It was stated $D(A)=W^{2,p}$. Could anyone give a proof of above claim? Here, \begin{equation*} D(A) :=\left\{v\in L^p(\mathbb{R}) : \lim_{t\to ...
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1answer
24 views

norms on vector space are equivalent iff they have same topology

Note that (1) all norms on finite dimensional vector space are equivalent. And here (2) norms on vector space are equivalent iff they have same topology. Question : How can we prove (2) ? Please ...
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45 views

Compact matrix integral operator bound via its kernel

Let $\mathcal{H} = L^p_{n}(0,a)$, where $p \in \mathbb{R}^+$, $p \geq 1$ and $n \in \mathbb{N}$, be the Hilbert space of vector-valued functions defined on the finite interval $(0,a) \subset ...
2
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1answer
48 views

Significance of having closed range for an operator?

One thing I think is of great importance is that it seems to correspond to possibilities similar to rank nullity theorem and further the fredholm alternative much as in finite dimension. Is this a ...
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1answer
62 views

Operator algebra generalization of linear algebra result on diagonalization of commuting operators with distinct eigenvalues

In linear algebra it is true that: a) if $\mathcal{A}$ is a set of unitarily diagonalizable matrices (in $\mathbb{C}$, i.e. normal matrices) that commute with each other then they are simultaneously ...
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22 views

Lower bound on multiplicative norm of Banach algebra.

Let $A$ be a Banach algebra over $\mathbb{C}$ and $N: A \to \mathbb{R}_{\ge 0}$ the corresponding multiplicative norm. For $a \in A$, do we have$$\limsup_{n \to \infty} (N(a^n))^{1\over{n}} \le ...
2
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0answers
72 views

Dual space $L^p$

Take a probability space $(\Omega,\mathscr{E},\mathbb{P}).$ Then it is known that $L^\infty \subset L^p \subset L^q \subset L^1$ for $\infty \ge p \ge q \ge 1.$ Let $l: L^p \rightarrow \mathbb{R}$ be ...
5
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1answer
106 views

Does an extension operator in Sobolev spaces commute with derivative operators?

Assume that $\Omega\subseteq \mathbb R^d$ is open and has a Lipschitz boundary. Let $\tau\geq0$. Then we know that there exists a linear operator $E:H^\tau(\Omega)\to H^\tau(\mathbb R^d)$ such that ...
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1answer
54 views

Antiderivative as an integral operator from $L^2(0,2\pi)$ to $L^2(0,2\pi)$

I am starting to study Functional Analysis on Hilbert Spaces and I am studying the following operator: $$T:L^2(0,2\pi) \rightarrow L^2(0,2\pi) $$ where $$Tf:(0,2\pi) \rightarrow \mathbb{R} \\ ...
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1answer
37 views

determinate moment problem

if the moment problem is determinate i.e given $L(f)= \int_{\mathbb{R}} fd\mu$ then $\mu$ is unique, how can i show that the the space of polynomials with complex coefficients i.e $C[x]$ is dense ...
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1answer
35 views

Why is the dual space with the weak*-topology a topological vector space?

In my lecture-notes on functional analysis I've found the fact that the dual-space $X^*$ with the weak*-topology of a real vector-space $X$ is a topological vector-space. I've tried to prove it, but ...
2
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1answer
41 views

Finding closure/interior of subset of function space

Consider the subset $$A=\left\{f\in C(\Bbb R): |f(x)|< \frac{1}{1+|x|} \, \text{for all } x\in \Bbb R\right\}\subset \left\{f\in C(\Bbb R): \lim_{|x|\to \infty}f(x)=0 \right\}=X.$$ where $X$ is ...
2
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1answer
94 views

Positive elements in * Algebras

If one has a * Algebra $W$ one can define the notion of a positive element via the spectrum $\sigma$: $A \in W$ is positive if $A^* = A$ and $\sigma(A) \subset \mathbb{R}_{≥0}$. If $W$ can be given ...
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1answer
84 views

The set of isomorphisms, $Iso(X,Y)$ is open.

Let $X,Y $ be Banach. I want to show that, $GL(X,Y) = \{ A\in L(X,Y), B \in L(Y,X) : BA= id_{x} \ \ \text{and} \ BA = id_{y} \} \subset^{open} L(X,Y)$ My attempt, Let $T \in Iso(X,Y), \ Q \in ...
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2answers
69 views

Hint for Lebesgue theory/functional analysis type of problem

I am trying to solve the following problem, but I am not too familiar with functional analysis. Could you guys tell me where I should start? Thanks! Let $f \in L^1(\mathbb{R})$ and define $$f_n(x) = ...
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1answer
73 views

How to prove the convergence in such a case?

$$ \dot x(t)=Ax(t)+Bu(t),\quad t\in[0,T]\\ x(0)=x_0\\ u\in L^\infty([0,T],\mathbb R^m)\text{ and }x\in L^\infty([0,T],\mathbb R^n) $$ It is known that $\lim_{a\to 0}||u_a-u_0||_{L^2}=0$, where ...
3
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1answer
62 views

Is $(\oplus\ell_2^n)_{\ell_1}$ complemented in $\ell_1\oplus_\infty\ell_p$?

Fix any $1<p\leq 2$. Let us recall that $E:=(\oplus\ell_2^n)_{\ell_1}$ is just the space of sequences $(x_n)_{n=1}^\infty$, $x_n\in\ell_2^n$, such that $(\|x_n\|_{\ell_2^n})_{n=1}^\infty\in ...
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2answers
47 views

Show that if T commutes with all M then T is a multiple of identity

Let $X$ be Banach and let T $\in \mathcal{L}(X)$ s.t $TM=MT$ $\forall $ M $\in \mathcal{L}(X)$. Then $T$ is a multiple of the identity. I did proof by contrapostition, but Im not 100% confident Im ...
2
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1answer
63 views

Inner product from Fourier-like kernel

A kernel $K\colon [0,1]^s\times [0,1]^s \rightarrow\mathbb{R}$ is a symmetric and positive semi-definite function (meaning that for any $v_1,\ldots,v_m\in [0,1]^s$ and any $m\geq 1$, the matrix ...
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1answer
49 views

two generalizations of Lax-Milgram theorem

Apart from the classic Lax-Milgram theorem which is powerful under the Hilbert setting, there are two generalizations: (from Wikipedia) Babuska-Lax-Milgram theorem suppose $U,V$ Hilbert spaces ...
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55 views

Why do these Integration-by-Parts Evaluation Terms Vanish?

The Associated Legendre operator is $$ L_mf = -\frac{d}{dx}\left((1-x^{2})\frac{df}{dx}\right)+\frac{m^{2}}{1-x^{2}}f, $$ where $m$ is a positive integer. For the purposes here, define ...
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31 views

$W = \text{Span}(V \cup\{y\})$ is a closed subspace if $V$ is. (In Hilbert Space) **Edited**

Let $V$ be a closed subspace of $\mathcal H$ and if $y \in \mathcal H$ but $y \notin V$ prove that $W = \text{Span}(V \cup\{y\})$ is a closed subspace. Here's my attempted proof. Proof: Note ...
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1answer
39 views

About the “Bounded Convergence Theorem”

The ``Bounded Convergence Theorem" states that "If a sequence $\{f_n\}$ of measurable functions is uniformly bounded and if $f_n \rightarrow f$ in measure then $lim_{n \rightarrow \infty } \int f_n ...
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37 views

Show homeomorphism of a map onto infinite product

Establish a map f: $\{0,1\}^\mathbb{N}$ $\to$ $\{0,1\}^\mathbb{N} \times \{0,1\}^\mathbb{N}$ , and let ($\{0,1\}^\mathbb{N}$, T) be the topological space, and T $\times$ T as the product topology for ...
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1answer
82 views

Showing $||T||_{op}$ is bounded

Let $X$ be a Banach space and $Y$ a normed vector space over the field $\mathbb{K}$. Let $F \subset L(X,Y)$ (where $L(X,Y) = \{{A ∈ Hom(X, Y) | A \ \text{continuous}} \} = \{A ∈ Hom(X, Y) \ | \ \ ...
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1answer
31 views

let X has a finite dimensional show that X is reflexive.

We know If a normed space X is reflexive, then $X'$ is reflexive.and also Reflexive normed spaces are Banach. but can you proof if X has a finite dimensional Then X is reflexive.
4
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1answer
73 views

Let $a_f=\text{ arg} \min_{a} \int \left|f(x)-a\right| dx$ and $a_g= \text{ arg} \min_{a} \int \left|g(x)-a\right| dx$, is $a_f \le a_g$?

Let $ f(x) \le g(x) $ and assume that $g(x),f(x) \in L^1$ let \begin{align} a_f= \text{ arg} \min_{a } \int_A \left|f(x)-a\right| dx\\ a_g=\text{ arg} \min_{a } \int_A \left|g(x)-a\right| dx ...
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0answers
24 views

The distance between two convex functions

I have a sequence $(u_t)\subset L^2(\Omega)$ so that the function $f(t)= \|u_t-u_0\|_{L^2}$ is continuous and strictly increasing and $f(0)=0$. Now, given $u'=u_0-v$ where $\int v =0$ and $\int v^2=c$ ...
1
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2answers
49 views

Can $\text{ arg}$ be thought of as operator?

Forgive me if the question is to vague. The argument, denoted by $\text{arg}$, is a commonly used notation. I am specifically interested in the following use of $\text{arg}$: \begin{align} a=\text{ ...
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1answer
56 views

Intersection of decreasing sequence of non empty closed sets

Let X is Banach space. Suppose $B_n$ are decreasing sequence of non empty closed balls. Prove their intersection is non-empty. I have some idea. Idea is pick $x_n \in B_n\backslash B_{n-1}$ in such ...
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0answers
16 views

Understanding the Quotient norm, constructed example

I just read the definition of the quotient norm, but it had no examples, so I constructed my own. Is this correct? Let $(X=\Bbb R^3,\|\cdot\|)$ with Euclidean norm and $Y\subset X$, $\quad ...