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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268 views

Semi-norms in Functional Analysis

I'm self-studying functional analysis. The following is from Rudin's "Functional Analysis, 2nd edition". It consists of parts from question 7 and 13 from the first chapter. I am not sure if my answers ...
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40 views

Norms and convergence in $\mathcal{C}^{\infty}(O)$

Let $O \in \mathbb{R}^d$ be open, $K \subseteq O$ compact and $n \in \mathbb{N}$. For $f \in \mathcal{C}^{\infty}(O)=\mathcal{E}(O)$ we define $$\|f\|_{n,K}^{(1)}:=\sup_{|\alpha| \leq n} \|D^{\alpha} ...
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72 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
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43 views

Bounded Functions: Uniform Closure

Let $E$ be a Banach space. Consider the space of bounded functions $\mathcal{B}:=\{f:\Omega\to E:f\text{ bounded}\}$ equipped with the supremum norm ...
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39 views

Linear operator determined by its action on a complete orthonormal sequence

Let $(e_n)$ be a complete orthonormal sequence in the Hilbert space H and let $(\lambda_n)$ be a sequence of scalars, not necessarily bounded. I'm trying to prove that the operator defined as $$Tv = ...
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66 views

what is the meaning of asphericity of convex set in a linear normed space?

I am trying to understand the definition of spherical to within $\epsilon$ for a convex body in a linear normed space, as given in Aryeh Dvoretzky's paper [1] (section 2, page 203): A convex set ...
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56 views

Need help considering series like these: $\sum_{n=1}^\infty\langle x,e_n\rangle e_n$

I'm working in a Hilbert space $H$ with ONB $(e_n)$ and I have $\alpha=(\alpha_n)\in\ell^\infty$. I have an operator that looks like this: $$T_\alpha x=\sum_{n=1}^{\infty}\alpha_n\langle ...
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60 views

A Particular Frechet Derivative and Interpretation

I would like find the Fréchet derivative of the following functional: $$ \begin{align} F : C[0,1] &\rightarrow \mathbb{R}\\ w &\mapsto \frac{\int_0^1 xw(x)f(x) \, dx}{\int_0^1 w(x)f(x) \, dx}. ...
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33 views

How to show $\mu_A(x)=\inf\{\alpha>0: \alpha^{-1} x\in A\}$ where $\mu_A(x)$ is the Minkowski functional of $A$?

Let $X$ be a $\mathbb K$-vector space ($\mathbb K=\mathbb R$ or $\mathbb C$) and suppose $A\subset X$ is convex (and absorbing). How to show $$\{x\in X: \mu_A(x)<1\}\subset A?$$ Above ...
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60 views

Polar decomposition in a von Neumann algebra

Let $M \subseteq B(H)$ be a von Neumann algebra and $T \in M$. If $T=U|T|$ is the polar decomposition of T, why is $U \in M$? I'm thinking it's because $M$ is SOT-closed, but I'm not entirely sure.
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44 views

unbounded self-adjoint operator

Given an operator $T:D_1(T)\subset L^2 \rightarrow L^2$ and the same operator $T:D_2(T) \subset L^2 \rightarrow L^2$, such that the operator is both times self-adjoint and closed, with $D_1(T) \subset ...
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49 views

The Nash inequality on a compact manifold without a boundary.

Let $M$ be a compact manifold without boundary. Does the Nash inequality $$\lVert u \rVert^{1+\frac 2n}_{L^2} \leq C\lVert u \rVert^{\frac 2n}_{L^1} \lVert \nabla u \rVert_{L^2}$$ or something ...
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87 views

Kadison's Inequality

Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional. Is there a simple proof for Kadison's inequality: $$|\omega(A)|^2\leq\|\omega\|\cdot\omega(A^*A)$$
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49 views

Unbounded Operators: Notation?

For continuous a.k.a bounded operators we have $\mathcal{B}(X,Y)$ stressing on boundedness and $\mathcal{L}(X,Y)$ stressing on linearity entailing $\mathcal{C}(X,Y)$. Is there a notation for ...
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119 views

Can I easily deduce this stronger spectral theorem from this weaker one?

I've just read a nice proof that: For $T$ a self-adjoint bounded linear operator on a Hilbert space $E$, there exists a unique $C^*$-algebra isomorphism $C(\sigma (T)) \rightarrow A_T$, from the ...
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286 views

Weak Derivative Heaviside function

I have to prove that the Heaviside function $$ H(x):=\begin{cases} 1 &\mbox{if } x \in [0,+\infty) \\ 0 &\mbox{otherwise}\end{cases} $$ doesn't admit weak derivative in ...
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291 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
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79 views

Kaplansky's Density Theorem for Unitary Operators

Let $M \subseteq B(H)$ be a *-subalgebra containing the identity on H. If there is a unitary T in the unit ball of the SOT-closure of $M$, is there a net of unitary oprators in the unit ball of $M$ ...
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2answers
45 views

Are bounded linear functionals on $L^{\infty}$ of “bounded variation?”

Let $(X,\mathscr M,\mu)$ be a measure space and let $L^{\infty}$ be the set of (equivalence classes of) essentially bounded measurable functions on it. Suppose that $\phi\in (L^{\infty})^*$; that is, ...
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41 views

$g(T)$ bounded implies $T$ bounded, if $T$ is linear and $g$ is bounded linear functional

Suppose $X$ is a Banach space and $y$ is a normed linear space and $T :X\to Y$ is linear map such that for every bounded linear functional $g\in Y^*$ we have $g( T)$ is bounded . Show that $T$ is ...
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29 views

Unique extension of operators

Let $p,q$ be conjugate exponents with $p\in(1,\infty]$ and consider a measure space $(X,\mathscr M,\mu)$. Suppose that $S:L^p\to L^p$ and $T:L^p\to L^p$ are both bounded, linear operators. Assume that ...
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36 views

norm of projection

How to find the norm of the following projection:( sup-norm) $c$ is the space of convergent sequences. $P:c\to c_0$ by ...
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69 views

Blind deconvolution of a function convolved with itself

I have a function/vector $f$ that I know is the result of an unknown function $g$ convolved with itself: $f = g \ast g$ Is there any way to do a blind deconvolution on $f$ with this constraint?
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26 views

Why do we have $\Vert y_n\Vert^2=\langle x;y_n\rangle$

I can't see why the following equalities are valid: Let $X$ be an inner product space with an othonormal system $e_n, n\in\mathbb{N}$ and $x\in X$. Define $y_n=\sum_{i=1}^n\langle ...
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2answers
77 views

Why unit open ball is open in norm topology, but not open in weak topology?

Why unit open ball is open in norm topology, but not open in weak topology? I will be grateful for any explanation. Edit: Obviously in infnite dimensional spaces.
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50 views

bounded sequence in $L^p(\mathbb{R}^n)$ that converges a.e.

Let $1<p<\infty$. Let $\{f_k\}$ be a sequence in $L^p(\mathbb{R}^n)$. Suppose $f_k\to f$ a.e. and there exists $C>0$ such that $||f_k||_p\leq C$ for all $k$. Prove that for all $g\in ...
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61 views

Convergence, Banach-Steinhaus

Let $(\phi_n)$ be a sequence of continuous linear functionals from $X$ - Banach space to $\mathbb{R}$ such that $(\phi_n(x))$ is convergent for all $x\in X$. Show that if a sequence $(x_n)$ is ...
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59 views

What is wrong with this “counterexample” of boundedness of weakly convergent sequences?

Weakly convergence sequences $\{u_n\}$ in a Hilbert space $H$ are bounded. Here is an attempted "counterexample". What is wrong with this? Let $H = \ell_2(\mathbb{N})$, and let $\{e_n\}$ be the ...
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45 views

Reference for $f \in L^{p,\infty} \cap L^{q}$ then $f \in L^r$ for $p < r \leq q$

Okay, so I think I've shown that if $f \in L^{p,\infty} \cap L^{q}$ with $p < q$ then $f \in L^r$ for $p < r \leq q$ where $L^{p, \infty}$ denotes the weak $L^p$ space. what I did was I wrote $$ ...
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77 views

Convergence weak* in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ implies convergence weak* a.e in $L^2(\mathbb{T}^2)$?

Suppose $x_n\rightharpoonup x$ in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ with the weak* topology, in other words, $\forall f\in L^{1}([0,T];L^2(\mathbb{T}^2))$ we have $$\lim_{n \to\infty} \int _0 ^T ...
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39 views

Is the complement of a set 1st category set $X$ of 2nd category?

If I have some metric space $M$ and find that $X \subset M$ is of 1st category (resp. 2nd category), is the complement of $X$, $X^c$ of second cateogry (resp. 1st category)? Thanks!
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27 views

Showing a composition is in $L^2$

Let $f$ be a smooth increasing function passing through the origin with $|f(x)| \leq C_1(1+|x|)$. Define $F(r) = \int_0^r f(s)\;ds$ and $F^*(t) = \sup_{r \in \mathbb{R}}tr - F(r).$ Note that ...
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25 views

The exactness of a C*-algebra

Here is a quotation: Corollary 3.7.12 If $\Gamma$ is a non-amenable residually finite group, then $C^{*}(\Gamma)$ is not exact. It follows from this corollary that $B(l^{2})$ is not exact ...
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73 views

If $X,Y$ are Banach spaces then for every $(y_n)\rightarrow 0 \exists (x_n)\rightarrow 0 : f(x_n)=y_n$

Let $X,Y$ be Banach spaces and $f:X \rightarrow Y$ be a linear, continuous and surjective map. Show that $\forall (y_n)\subset Y, \:\: y_n \rightarrow0 \:\:\: \exists (x_n) \subset X ...
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195 views

Weak convergence of continuous functions

Let $X$ be an LCH space and $C_0(X)$ the set of continuous vanishing functions on $X$. If $C_0(X)$ is given the structure of a Banach space with the sup-norm, then its weak topology is given by the ...
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38 views

Verification of conclusions regarding duality maps

I have two conclusions drawn from two results. I want to know how valid these two conclusions are. Firstly Consider the duality mapping(set-valued) $J:X \rightrightarrows X^{*}$ defined: $J(u) := \{ ...
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82 views

Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$

Let $(x_n)$ be weakly convergent, but not norm convergent, sequence from a Banach space. Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$. Any help?
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120 views

Weak* compactness of the unit ball

Things that we know: In any topological space compactness implies sequential compactness If E is any topological space the then the closed unit ball $$ B_E=\{f\in E^*; \|f\|\leq 1\} $$ is compact ...
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73 views

Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
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24 views

The norm on tensor product

Here is a quotation of a book: Let $B, ~C$ be unital C*-algebras and $A$ be a nonunital C*-algebra, $\|\cdot\|_{\alpha}$ be a C*-norm on $B\odot C$ (the tensor product) and $\|\cdot\|_{\beta}$ be ...
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49 views

Span of a closed subspace

Let $Y$ be a closed set of a Banach Space $X$. Is it true that the linear Span($Y$)is also closed? For the examples I have tried, I see that the result holds true. I understand that the linear span ...
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29 views

Exact sequence of tensor product

Here is a quotation of a book: Proposition 3.7.1. If $0 \rightarrow J \rightarrow A \rightarrow (A/J)\rightarrow 0$ is an exact sequence, then for every $B$, the natural sequence $$0 \rightarrow ...
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129 views

What is the dual space in the strong operator topology?

Let $X$ be a Banach space, the strong operator topology on the space of bounded linear operators $\mathcal{B}(X)$ is defined by the family of continuous semi-norms $A\to\|Ax\|$, $x\in X$. What is the ...
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77 views

Decay of a Convolution

Let $f, g \in L^1\cap L^\infty(\mathbb{R}^d)$ be probability distributions on $\mathbb{R}^d$, and suppose at large $|x|$, $f$ decays like $|x|^{-\alpha}$ while $g$ decays like $|x|^{-\beta}$, with ...
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38 views

Dense subspace of linear space and functional equal to $0$

Let $X$ be a normed space over field $\mathbb{K}$ and let $X_0$ be a linear subspace of $X$. I have to prove that: $X=\overline X_0 \iff$ For every linear continuous functional $\phi : X\rightarrow ...
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45 views

How to show whether this operator is normal? self-adjoint? unitary?

Consider the Hilbert space $H=l_2$ over the complex field and $A:H\rightarrow H.$ How to show whether this operator is normal? self-adjoint? unitary? ...
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27 views

How to show that the spectrum is equal to the range of $y$

How to show that the spectrum of $T_y$ is equal to the range of $y$ Given $y\in C[0,1]$ and $T_y: C[0,1] \rightarrow C[0,1]: x\mapsto x\cdot y$ Any help is appreciated, thanks.
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37 views

questions about $L^p$ space with $0<p\leq 1$ parallel to the case $1<p$

Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete. Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with ...
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28 views

Unbounded spectrum can be empty

Can someone please provide an example of a Hilbert space $H$ together with a (prefer densely defined if possible) unbounded operator $T$ such that $\sigma(T)$ is the empty set? I have tried to use ...
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46 views

Small question about strong convergence

I have a small question I have that $ \lambda_1 ||v_0||^2_{L^2(0,1)}=||v_0||^2_{H^1_0(0,1)}$ and that $v_n\rightarrow v_0$ on $L^2$ (strongly) From the Poicaré inequality i have that ...