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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78 views

Prove that a set is dense and of the first category in $L^2(T)$

Define the Fourier coefficients $\hat{f}(n)$ of a function $f\in L^2(T)$, $T$ is a unit circle, by: $\hat{f}(n)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(e^{i\theta})e^{-in\theta}d\theta$ Let ...
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1answer
64 views

Is this a separable Banach space?

Define $X=\{ f\in C^1 ( \mathbb{R} ):~\int_{-\infty}^{\infty} \left| f'(x) \right| \mbox{d}x < \infty \}$ with norm $\| f \| = \left| f(0)\right| + \int_{-\infty}^{\infty} \left| f'(x) \right| ...
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35 views

Extending operator from subspace of Banach space to whole space

Let $Y$ be subspace of Banach space $X$. $T:Y \rightarrow \ell_{\infty}$ is continuous linear operator. Show that there is linear function $S:~X \rightarrow \ell_{\infty}$ such that $S\vert _{Y}=T$ ...
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1answer
78 views

Measurability of composition (Bochner function)

Let $g \in L^2(0,T;X)$ and $f:[0,T]\times X \to \mathbb{R}$ is such that $$\frac{d}{dt}F(t,x) = f(t,x).$$ so $t \mapsto f(t,x)$ is measurable on $[0,T]$ (but I don't know in what sense) for fixed $x$. ...
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0answers
44 views

What is the motivation for “continuity in the sense of distributions”?

Let $M$ be a compact (real) manifold and let $\Omega^m_c(M)$ be the compactly supported $m$-forms on $M$. Apparently a linear map $T : \Omega^m_c(M) \to \mathbb{R}$ is continuous "in the sense of ...
2
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1answer
181 views

Finding the adjoint for a diagonal operator in $\ell^2$.

Let $\{e_n\}_{n=1}^{\infty}$ be an orthonormal basis of the complex Hilbert space $\ell^2$. Fix complex numbers $\lambda_1,\lambda_2,\lambda_3,\dots$, let $$ \mathscr{D}(T)=\{\sum_{n=1}^{\infty} x_n ...
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2answers
63 views

Show that the closures in the topology weak and the norm are coincide:

Let $X$ be a Banach space and $Y \subset X$a vector subspace. Let $Y_f$ and $Y_F$ are the closures of $Y$ in the topology weak and the norm, respectively. Prove that $Y_F = Y_f$.
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113 views

Is the product rule true in a Banach algebra?

Let $X$ be a Banach space and $\mathcal{L}(X)$ the Banach algebra of all bounded linear operators $L:X\to X$, where the norm is given by $$\|L\|_\mathcal{L}=\sup\{\|L(x)\|_X;\;\|x\|_X=1\}$$ and the ...
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1answer
83 views

Hilbert space, orthonormal system, compact set of vectors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...
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2answers
87 views

Dolbeault cohomology and analytic regularity

Let $M$ be a complex analytic $n$-manifold. The Dolbeault cohomology complex is defined using a quotient space of smooth differential forms. My question is : would it make a big difference if we were ...
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1answer
110 views

A matrix in $SL(2)$ has it's supremum norm and infimum fulfilled by orthogonal vectors

I am having trouble proving the next statement: If $B\in SL(2)$ and $||B||\neq 1$, for $||B||:= \underset{x\neq 0}{\sup}\big\{\frac{||B(x)||}{||x||} \big\} $, where $||\cdot||$ is the euclidian norm, ...
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1answer
49 views

a.e. differentiable + continuous implies sobolev function?

Let $f\in C^0(\Omega)$ (where $\Omega$ is a bounded Lipschitz domain in $\mathbb R^n$). Suppose that $f$ is almost everywhere differentiable (in the classical sense). Is this condition sufficient to ...
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1answer
44 views

Isometry from closed operator

I have a following problem Let $H$ be a Hilbert space. We have a closed densely defined operator $A \colon D \subset H \rightarrow H$, we know that $\|Ax\| = \|x\|$ for all $x \in D$, can we extend ...
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1answer
59 views

Compactness in $\mathbb{R}^{X}$

I'm reading a book chapter on weak topology, where the author identified the collection of all real functions on an abstract space $X$ with $\mathbb{R}^{X}$. I find it difficult to make sense out of ...
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40 views

Rotation semigroup and identity element.

Let $\Gamma =\{z\in\mathbb{C}:|z|=1\}$, and $X=C(\Gamma)$. The rotation semigroup $\{T(t)\}_{t\geq 0}$, is defined as $$T(t)f(z)=f(\mathrm{e}^{it}z),\quad f\in X.$$ $Z\in X$, s.t. for all ...
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1answer
46 views

Operator from $\ell_{4}$ to $\ell_{1}$ is compact, if it's continuous.

Define $T: \ell_{4} \rightarrow \ell_{1}$ as $Tx=(a_1x_1, a_2x_2, \ldots)$. I showed that $T$ is continuous if and only if $\sum \left| a_i \right|^{\frac{4}{3}} < \infty$. How can I prove that if ...
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1answer
37 views

Is operator from $X^{*}$ to $c_{0}$ compact

$X$ - Banach space. Let $X \ni (x_n)$ be such that $x_n \stackrel{\omega}{\rightarrow}0 $ weakly. Define $T: X^{*} \rightarrow c_0$ by $T=\left(x^{*}(x_n) \right)_{n=1}^{\infty}$. Is operator $T$ ...
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3answers
626 views

Compact operator with closed range has finite dimensional range

Let $X,Y$ be Banach Spaces, and let $T\in K(X,Y)$ be a compact operator from $X$ to $Y$. I have to prove that $T(X)$ is closed in Y if, and only if, $\dim(T(X))<\infty$. Can anybody help me with ...
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1answer
172 views

The proof that every bounded linear operator generates an unique uniformly continuous semigroup.

Let $X$ be a Banach space and $A: X \to X$ a bounded linear operator. So, $A$ is the infinitesimal generator of an uniformly continuous semigroup $\{T(t)\}_{t\geq 0}$ on $X$. The proof, as presented ...
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2answers
78 views

The space $\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$

For a open Lipschitz domain $\Omega$, consider the space $$A =\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}.$$ Now I heard somewhere that all the second derivatives of a function $u$ are ...
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1answer
94 views

Preannihilator of the image of an adjoint of a bounded operator

Let $E,F$ be normed spaces and $F\colon E\rightarrow F$ be a linear bounded operator. Denote by $$A'\colon F'\rightarrow E'$$ the adjoint of the operator between the topological duals of the normed ...
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1answer
82 views

Understanding this inner product

I want to find out under which conditions on $w$, we have that $$\langle f,g \rangle :=\int_0^1 f(x)\bar{g}(x)w(x) dx $$ a dot product?, where $f,g \in C([0,1],\mathbb{C})$ and $w \in ...
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1answer
74 views

Constructing Unbounded Linear Maps

Suppose that $X$ is an infinite-dimensional normed vector space over $\mathbb C$ (that is, the cardinality of any of its Hamel bases is infinite) and let $Y$ be another, nontrivial normed vector space ...
3
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1answer
91 views

How should I think about reflexive spaces? What “property” do I get from a space being reflexive?

Let $(X,\| \cdot \|_X)$ be a $\mathbb{R}$-vector space with bidual space $X^{**}$. We defined $X$ to be reflexive, if the canonical embedding $\mathcal I: X \to X^{**}$ with $$\mathcal I x(l) := ...
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1answer
37 views

Show: $\exists n\in\mathbb{N}$ so that $T^n\colon C([a,b])\to C([a,b])$ is $q$-contractive

Consider $-\infty<a<b< +\infty, F\in C([a,b]^2\times\mathbb{R}), f\in C([a,b])$ with $$ \exists L\geq 0~\forall x,y\in[a,b]~\forall u,v\in\mathbb{R}:~~\lvert ...
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2answers
481 views

Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces

In all books that I have checked the spectral theorem (every self-adjoint operator on a Hilbert space is unitary equivalent to a multiplication operator on some $L_2(\mu)$) is only stated for complex ...
3
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1answer
146 views

Proof of a property of the uniformly continuous semigroups.

Let $X$ be a Banach space and $T:=\{T(t)\}_{t\geq 0}$ an uniformly continuous semigroup of bounded linear operators on $X$. So, we know that (i) $T(t)$ is linear and bounded for all $t\geq 0$; ...
3
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1answer
119 views

closed strong vs. closed in weak convergence

Studying a chapter about weak topologies and weak convergence I though the following which I have no idea how to prove or disprove it. So here it is: Question: Does there exist Banach space $X$ ...
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1answer
135 views

absolutely convergent series in Hilbert space

Is it possible to find an infinite dimensional Hilbert space, where every convergent series is absolutely convergent? I could not find any clue to find an example of such type or to disprove that. ...
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2answers
386 views

Collecting things that are preserved by (isometric) isomorphisms between normed spaces

I would like to collect a list of things that are preserved under isomorphisms and isometric isomorphisms. The reason is that I hope to get a better perception of their importance in Functional ...
2
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1answer
108 views

Hilbert Space and Projections

If $M$ is a closed subspace of the Hilbert space $H$ and $x$ $\in$ $H$, prove that: $$\underset{y \in M}{\min} ||x-y|| =\max\{|\langle x,z\rangle|:z \in M^{\perp}, ||z||=1\}.$$ There isn't a ...
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1answer
157 views

linear map of bounded sets into bounded sets implies a bounded operator

I was watching a video lecture on bounded linear operators from one normed linear space to another. It was stated that if $T$ sends bonded sets in $X$ to bounded sets in $Y$ then $T$ is a bounded ...
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1answer
224 views

Resolvent R(1) of the Laplace operator not compact

I want to show that $$R_\Delta(1):=(1-\Delta)^{-1} $$ is not compact in $\mathbb{R}^3$. I have found that for $\chi_{B}$ being the characteristic function for a set $B\subset\mathbb{R}^n$, ...
3
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2answers
70 views

Find a sequence of positive functions with non-trivial properties in $L^1([-\pi,\pi])$ and in $L^2([-\pi,\pi])$

I was asked to exhibit a sequence of positive functions $\{f_n\}_{n\in\mathbb{N}}$ belonging to $L^2([-\pi,\pi])$ such that: $\{f_n\}_{n\in\mathbb{N}}$ is strongly converging to $0$ in ...
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1answer
27 views

Countably additive function

Let R be a sigma algebra, and g be a real valued function on R such that for a sequence ($A_{n}$) of disjoint members of R, we have that g($A$) (where $A$ is the countable union) is equal to the ...
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1answer
46 views

Easy question about $C_c^\infty(0,T)$ and $C_c^\infty((0,T);X)$

Let $f \in C_c^\infty(0,T).$ It follows that $f \in C^k(0,T)$ for all $k$, and so if $t_n \to t$ then $$|f(t_n) - f(t)| + |f'(t_n) - f'(t)| + ... +|f^{(k)}(t_n) - f^{(k)}(t)| \to 0$$ for all $k$. Now ...
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2answers
129 views

Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
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0answers
52 views

weakly convergence of periodic functions

Let $ \displaystyle I \subset \mathbb R$ be a bounded interval and $\displaystyle 1 < p \leq \infty $. Let $ f \displaystyle \in L^\infty (\mathbb R) $ periodic function with $\displaystyle ...
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1answer
46 views

Translation is continuous

Let $\mathcal D$ be the space of 'test-functions'. Those are infinitely differentiable functions with compact support. Define the following convergence on $\mathcal D$. $(\phi_j) \to \phi$ in ...
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1answer
66 views

Supremum on $[0,T]\times A$ for $A$ compact

Let $A$ be a compact set in $\mathbb{R}^n.$ Let $f:[0,T]\times A \to \mathbb{R}$ be a function. Is it true that $$\sup_{t \in [0,T]} \lVert f(t) \rVert_{C(A)} = \lVert f\rVert_{C([0,T]\times A)}?$$ ...
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0answers
81 views

Tricky weak-* convergence question

Let $K$ be a compact (let's say, $K=[0,1]$ to be concrete) and let $\mu_n$ be a sequence of Radon measures converging weakly-* to another Radon measure $\mu$ on $K$. Let $h : K \rightarrow \mathbb{R}$ ...
2
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1answer
95 views

Convergence in Sobolev Spaces

Consider the bounded mapping $A:W^{1,p}(\Omega) \rightarrow W^{1,p}(\Omega)^{*}$ where $A$ is defined as: $\langle A(u),v \rangle\text{ } := \int_{\Omega}a(x,u,\nabla u)\cdot \nabla v + c(x,u,\nabla ...
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1answer
75 views

Is this an error by my lecturer? Closure definition

My lecturer defined the closure of a set $M$ to be $\overline{M}=\bigcap \{ F \mid F$ is closed and $F\supseteq M \}$. However, in other modules it has been defined as $\overline{M}=\bigcap\limits_{F ...
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3answers
84 views

For closed sets, is $\text{cl}(A+B)=\text{cl}(\text{cl}(A)+\text{cl}(B))$?

Let $A$ and $B$ be nonempty subsets of $\mathbb{R}^n$, then is $\text{cl}(A+B)$ equal to $\text{cl}(\text{cl}(A)+\text{cl}(B))$? If that is true, then how to prove it? If they are not equal, then ...
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1answer
63 views

Cross Product Algebras references

Can someone give some references to introductory books or online notes about group algebras and cross-product algebras ? I've already searched on Google (but only for some online notes). The purpose ...
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0answers
57 views

properties of integral operator $x^{-1}\int_0^xf(x,y)v(y)dy $

here we have two cases to study $(1)$ let us fix any $f \in C^{1}[ [0,1] \times [0,1]]$ ($k \neq 0$). Set $$[T(v)](x) := x^{-1}\int_0^xf(x,y)v(y)dy $$ for any $x \neq 0$ otherwise $[T(v)](0) := ...
0
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0answers
102 views

Absolutely convergent series in normed linear space

I want to prove that in a normed linear space $X$ if for all absolutely convergent series $\sum\limits^{\infty}_{n=1}x_n$, the series $\sum\limits^{\infty}_{n=1}T(x_n)$ is convergent, then $T:X\to Y$ ...
2
votes
0answers
58 views

Exploiting the compactness of the unit circle to prove the following proposition.

I am trying to prove that a locally convex topological vector space is equivalent to a semi-normed topological vector space. I have worked through the proof but I am unsure because of the following ...
1
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1answer
74 views

A problem of functional analysis

I'm studying functional analysis and I'm having a "lot of problem with this problem". The question is the following: Let $E$ be a normed space. Is it true that for all countinuous map ...
0
votes
1answer
45 views

What is the argument for continuity here

Here is a paragraph in Murphy's book on operator theory: I don't understand how continuity of $z \mapsto \tau_z$ follows from the continuity of $z \mapsto \tau_z(f)$. If $f \circ g$ is continuous ...