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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Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
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2answers
37 views

An equality in Hilbert spaces

To understand a proof in functional analysis I need to understand why the following equation is true: $$\lVert x\rVert^2 - \sum_{j=1}^n |x_i|^2 = \Biggl\lVert x-\sum_{i=1}^nx_ie_i\Biggr\rVert^2$$ ...
3
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1answer
11 views

Limit (in probability) of sequence of independent random variables

We have $\{X_n\}$ independent random variables convergent to $X$ in probability. I was aked to prove that X is constant, but I can only do that when some $X_n$ has finite variation or what is in fact ...
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21 views

How to get the potential and the gradient of this function?

How to calculate the potential $P_3:\mathbb{R}^3\rightarrow\mathbb{R}$ with the $\nabla P_3=f_3$ of the function $f_3:\mathbb{R}^3\rightarrow\mathbb{R}^3$ with $\large ...
2
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1answer
18 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta( \frac{x}{h}).$$ This ...
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1answer
28 views

What is a linear functional on continuous functions on the real line not given by a measure?

What is a positive linear functional on continuous functions on the real line not given by integration against a measure? I know that the dual of $C_c(\mathbb R)$ is the set of Radon measures, ...
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1answer
16 views

Reference for the statement “bilinear form $a$ is symmetric if and only if the operator $S$ is self-adjoint”

Thanks to Riesz representation theorem, a continues bilinear (sesquilinear) form on Hilbert space $$a: \mathcal H\times \mathcal H\rightarrow\mathbb R \ \ (\text{or} \ \ \mathbb C)$$ can be ...
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2answers
59 views

Are continuous functions with compact support bounded?

While studying measure theory I came across the following fact: $\mathcal{K}(X) \subset C_b(X)$ (meaning the continuous functions with compact support are a subset of the bounded continuous ...
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4 views

How to construct a fundamental solutions of a PDE from well-posedness?

A fundamental solution of a linear operator $P$ on a manifold $M$ is a distribution $G$ such that: $$P(G)=\delta(x-y)$$ In formal terms this is stated as given a test function $\phi$ then: ...
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0answers
10 views

Question about sub differentiability of convex function

I am reading this book to study sub differentiability. On page 20 it says, as $V$ is a normed linear space for simplification, "a continuous affine function $l(v)$: $V\to \mathbb R$ everywhere less ...
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0answers
6 views

Question about affine isometric action

Recently, I read the book Kazhdan's Property (T). There is a lemma on the page 75 (Lemma 2.2.1) as following: Lemma. Let $\pi$ be an orthogonal representation of $G$ on $H^0$. For a mapping $\alpha: ...
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29 views

Is $H^2(\Omega)\cap H_0^1(\Omega)$ compactly embedded on $H_0^1(\Omega)$?

Considering $\Omega$ bounded and $\partial \Omega$ smooth. I already know that $H^2(\Omega)\cap H_0^1(\Omega)$ is continuously embedded on $H_0^1(\Omega)$, thus if I take a bounded sequence in ...
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0answers
16 views

Conway’s Functional Analysis, VIII §3 Exercise 11

This exercise is a step to proving inequalities involving non-commuting elements of a C*-algebra. (In particular in the subsequent exercise 12). Unfortunately I do not see, how to prove part (a): For ...
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0answers
22 views

Dimension of a measure in terms of linear form on continuous functions

So this might be a bit of a weird question, but here goes. It is well-known (Riesz representation theorem) that the dual space of continuous functions on a compact $K$ identifies with the space of ...
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0answers
22 views

Prove or disprove that $φ_v:u\mapsto \langle\mathcal A u,v\rangle$ is in $V^*$

Let us consider a linear and continuous operator on a Hilbert space $V$, $\mathcal A:V\rightarrow V$, such that: $$\|\mathcal A u\|\leq M \|u\|, \ \ \forall u\in V, M>0$$ and now consider ...
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1answer
29 views

Example: Operator with empty spectrum

I tried Google and a few books but couldn't find a suitable example. Does anyone know an example of an (unbounded closed) Operator BETWEEN HILBERTSPACES(!), that has empty spectrum? Thanks for your ...
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2answers
26 views

Compact sets are bounded: shape of the cover matters?

To prove a compact sets is bounded, we assume there's a "open ball cover" (each with R=1) that covers the set. And take maximum distance over the center of the balls +2 as the boundary. Why could we ...
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1answer
34 views

Continuous linear operator T at a point T is then continued

The problem is the next If T is continuous at a single point, it is continuous, without using that T is continuous iff T is bounded. I tried this result as follows If T is continuous at a single ...
3
votes
1answer
35 views

Strict positiveness on a C*-algebra given by generators and relations.

Let $A$ be a C*-algebra with generators $a_1,a_2,\ldots,a_n$ and some (non-important) relations (the relations imply that $\|a_i\|\leq 1$, so that $A$ exists). Among the given relations we have that ...
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0answers
36 views

A convergence problem for a sequence of functions

Let $\{f_n\}$ be a sequence of functions in $L^p$ with $0<p<1$ let $b>1$ and $(b^n(f_{n+1}-f_{n})\rightarrow 0)$ what we can say about $\{f_n\}$ ?is it Cauchy?
3
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1answer
34 views

Estimate for a weak solution to a PDE

Let $f \in L^2(B_R(0))$ and let $u \in W^{1,2}(B_R(0))$ be a weak solution of the equation $$Lu = - \sum_{i,j=1}^{n} D_i(a_{ij}D_ju)+ \sum_{i=1}^{n} b_i D_i u + cu =f.$$ There are constants $0 \le ...
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1answer
19 views

Codimension 1 closed subspace as a kernel of a functional

My non-linear analysis book says that if I have a linear operator $T:X\to Y$ with close range $R$ and $\operatorname{codim}(R)=1$ (and also $\dim(\ker(T))=1$) then there exists $\phi\in Y^{*}$ such ...
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0answers
21 views

Why does $M$ have a limit rank in the operator norm? Why is $S$ bounded? [on hold]

Define operator $S$ and $M$ on $\ell^2$ by $(SX)_n = \begin{cases} 0 & n = 0 \\ x_{n - 1} & n > 1 \end{cases}$ $(Mx)_n = \dfrac{1}{n + 1} x_n,\qquad n \ge 0$ Why does $M$ have a limit ...
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1answer
33 views

Sobolev space on a closed subset

Hi I have a question about Sobolev spaces. Let $U \subset \mathbb{R}^{d} $ be a bounded open subset and $dx$ be a Lebesgue measure on $U$ \begin{align} W^{1,2}(U):=\left\{u \in L^{2}(U;dx): ...
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votes
1answer
16 views

Infinite sum of bounded linear operators on a Hilbert space

Let $\mathcal{H}$ be an infinite-dimensional, separable, complex Hilbert space, and let $\mathbf{a}$ and $\mathbf{b}$ be bounded linear operators on $\mathcal{H}$ such that ...
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votes
2answers
26 views

Locally boundedness of some L^p spaces.

It is well known that $L^p$ spaces for $ 0<p<1 $ are not locally convex. I would like to know whether they are locally bounded.
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1answer
41 views

Show that the space $ℓ^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0\text{ for } j>>1\}$ is not complete

Show that the space $$\ell^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0 \text{ for } j\gg1\}$$ with inner product $$(a,b) \in ℓ^0\timesℓ^0 \mapsto \langle a,b\rangle =\sum_{j=1}^\infty ...
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1answer
25 views

Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...
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18 views

Domain of closed unbounded operator

Let $A$, $B$ be two closed unbounded operators such that: (1) there exists dense subspace $\mathcal{D}$ of $Dom(B)$ which is contained in $Dom(A)$, (2) for every $\psi \in\mathcal{D}$ it holds $$ ...
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votes
2answers
45 views

Differentiating $f(x)=\sum_{i=1}^{N}|x-y_i|^2$ where $y_1,…,y_N\in \Bbb{R}^n$.

Let $y_1,...,y_N\in \Bbb{R}^n$ and let $f(x)=\sum_{i=1}^{N}|x-y_i|^2$. I need to show that $f$ has a minimum. I try to differentiate but I am having troubles doing so. First of all, does $|x-y_i|$ ...
2
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2answers
36 views

Functional Analysis (Topological and Isometric Isomorphisms)

Give an example that if two normed linear spaces are topologically isomorphic then they need not be isometrically isomorphic. I searched my book and on the Internet but in vain.
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1answer
59 views

In the Hahn-Banach theorem, what is the purpose of the 'dominating function'?

I am studying functional analysis by reading "Elements of Functional Analysis" by IJ Maddox (which was the set text for the Open University's now discontinued course on this subject). In the ...
2
votes
1answer
27 views

Existence of the continuous spectrum of a possibly-unbounded, linear self-adjoint operator on a complex Hilbert space

Let $\mathbf{A}$ be a possibly-unbounded, linear self-adjoint operator on an infinte-dimensional, complex separable Hilbert space $\mathcal{H}$, and suppose we know the matrix elements $\langle ...
2
votes
1answer
27 views

The 1-Norm on a Quantum Group as a Supremum

To this MO question, Yemon Choi comments that If $\tau$ is a faithful normal trace on a von Neumann algebra $M$, then IIRC $\tau(|x|)$ is equal to the supremum of $|\tau(xy)|$ as $y$ runs over all ...
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1answer
19 views

application of positive linear functionl

The space $L^{1}(\mathbb R)$ is a commutative Banach algebra under convolution. A linear functional $F$ on $L^{1}(\mathbb R)$ is positive if $F(f^*f)\geq 0$ for $f\in L^{1}(\mathbb R).$ (where ...
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votes
0answers
7 views

Isometry and topological isomorphism produces a equivalence relation on the collection of all normed linear space over some field K

The concept of Isometry and topological isomorphism produces a equivalence relation on the collection of all normed linear space over some field K . I want to know how to proceed... Help needed
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6 views

Equivalence relation in the collection of all normed linear space over some field K.

The concept of equivalent norm produce an equivalence relation in the collection of all normed linear space over some field K. I've no idea how to make a start... Please help
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votes
1answer
34 views

$C^*\!$-algebra-normal element, self-adjoint element and spectrum [on hold]

Let $A$ be a $C^*\!$-algebra. Suppose $x$ is a normal element of $A$ and $\operatorname{spect}(x)$ lies in $\mathbb{R}$. Prove that $x$ is self-adjoint.
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0answers
21 views

If for every $a > 0$, $u \in C^\infty([a,\infty))$, then is $u \in C^\infty((0,\infty))$?

Suppose that for every $a > 0$, $u \in C^\infty([a,\infty))$. Does this imply that $u \in C^\infty((0,\infty))$? I think it is true when we just work in $C^0$, but with $C^\infty$ you need to ...
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39 views

Functional Analysis (Normed Linear Spaces)

Since every normed linear space is a vector space but every vector space is not necessarily a normed linear space. Give an example to show that a vector space is not a normed linear space that is norm ...
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1answer
31 views

$L_{P}[0,1]$ space for $0<P<1$ is metric space. [on hold]

For $0<P<1$, let $L_{P}[0,1]$ be the set of measurable functions $f : [0,1]\rightarrow R$ such that $\int{|f(x)|}^{p}dx<\infty$. How the function $d(f; g) =\int{|f(x)-g(x)|}^{p}dx$ is a ...
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1answer
23 views

$f(x,y,z)=ax+by+cz$. If $\mathbb R^3$ equipped with sup norm is f be bounded? If so find $\Vert f\Vert.$

It's very easy to see $f$ is bounded with respect to 2-norm which I've already done. $$|f(x,y,z)|\leq|a||x|+|b||y|+|c||z|$$ $$\leq\sqrt{a^2+b^2+c^2}\Vert(x, y, z)\Vert.$$ Then $\Vert ...
1
vote
1answer
19 views

Series Test for Integrability via the Distribition Function

I imagine that the following question has a well known (and perhaps, easily obtainable) answer, but I can't find it by myself nor along the references that I have in mind so far. So, if $f$ is a ...
0
votes
1answer
29 views

Integral operator

Let T: $C[0,1]\rightarrow C[0,1]$ be defined by $y(t)=\int_{0}^{t}x(\tau)d\tau$. Find Img(T). I know that Img(T)={$w\in C[0,1]:w=(Ty)(t) \text{ for some } t\in C[0,1]$}. Could you give me any ...
3
votes
1answer
57 views

Is the supremum norm the only $ C^{*} $-norm on $ {C_{c}}(X) $, equipped with the usual pointwise operations?

Let $ X $ be a locally compact Hausdorff space. Then $ {C_{c}}(X) $ is a commutative $ * $-algebra with respect to addition, multiplication, scalar multiplication and conjugation (all pointwise ...
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0answers
21 views

If $B$ is the closed unit ball of $X$, Why does $\varphi (B)$ is $\sigma $-dense in the closed unit ball of ${X^{**}}$?

Let $\varphi $ be the embedding of $X$ into ${X^{**}}$ . Let $\tau $ be the weak topology of $X$, and let $\sigma $ be the $weak^*$-topology of ${X^{**}}$--the one induced by $X^*$. If $B$ is the ...
0
votes
1answer
26 views

When the Multiplier algebra of a Banach algebra is exactly equal to the operator algebra?

Let A be a Banach algebra. B(A) and M(A) be the operator algebra and the multiplier algebra of A, respectively. When we have M(A)=B(A)?
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22 views

Banach space and Hamel Basis cardinality

No infinite-dimensional normed linear space with a Hamel basis having cardinality strictly less than c can be complete. Can we prove it without using AC or Hahn-Banach Theorem?
3
votes
0answers
35 views

Equivalence of norms in $C^1[0,1]$

i have the following problem/questions: I have to prove that $\lVert \cdot \rVert_1 \sim \lVert \cdot \rVert_{*} $ in $C^1[0,1]$; Where $\lVert \cdot \rVert_1$ is the usual $C^1[0,1]$ norm and ...
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0answers
26 views

Part of Lomonosov's Invariant Subspace Theorem

Let $X$ be a complex Banach space of infinite dimension, let $T\in\mathcal{B}(X)\backslash\{0\}$ be compact. Define $$\Gamma := \{S\in\mathcal{B}(X)\,|\,S\circ T=T\circ S\}$$and define, for each ...