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, ...

learn more… | top users | synonyms (1)

0
votes
0answers
9 views

The image of $H^2(\Omega) \cap H^1_0(\Omega)$ under $\partial_{x_i}$

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary. Let $A=H^2(\Omega) \cap H^1_0(\Omega)$. We have that $\partial_{x_i}(A) \subseteq B$ where $B=\{ f \in H^1(\Omega) : ...
1
vote
1answer
12 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 form on Hilbert space $$a: \mathcal H\times \mathcal H\rightarrow\mathbb R \ \ (\text{or} \ \ \mathbb C)$$ can be represented by a linear ...
-1
votes
2answers
31 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 ...
0
votes
0answers
2 views

Is this a valid construction for fundamental solutions 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: ...
0
votes
0answers
4 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 ...
0
votes
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: ...
0
votes
0answers
22 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 ...
0
votes
0answers
11 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 ...
1
vote
0answers
14 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 ...
0
votes
0answers
21 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 ...
2
votes
1answer
24 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 ...
1
vote
2answers
25 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 ...
0
votes
1answer
33 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
33 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 ...
1
vote
0answers
35 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
votes
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 ...
1
vote
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 ...
-2
votes
0answers
19 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 ...
2
votes
1answer
32 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): ...
0
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 ...
-2
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.
0
votes
1answer
40 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 ...
-1
votes
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 ...
1
vote
0answers
17 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 $$ ...
0
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
votes
2answers
33 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.
4
votes
1answer
58 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
26 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 ...
1
vote
1answer
18 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 ...
0
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
0
votes
0answers
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
0
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.
0
votes
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 ...
1
vote
0answers
37 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 ...
-1
votes
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 ...
0
votes
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 ...
2
votes
1answer
55 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 ...
0
votes
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
21 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)?
0
votes
0answers
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
34 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 ...
1
vote
0answers
24 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 ...
2
votes
1answer
47 views

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$. How will we prove the converse implication. One sided implication for Hilbert Space is proved in ...
1
vote
1answer
37 views

Basis for $l^{\infty}$

As the question stated, we know that $\{e_i\}$ doesn't form a basis for $l^{\infty}$. So how can we find a basis for $l^{\infty}$, no matter it is Schauder or Hamel basis.
0
votes
1answer
28 views

The Legendre-Fenchel transform of $BV$ semi-norm

I am reading a numerical paper in which it calls some "easy" facts from convex analysis but I can't justify it... Let $\Omega\subset \mathbb R^2$ be open. Define for a function $u\in L^1(\Omega)$ and ...
1
vote
1answer
39 views

existence of functionals

Let $X$ be a finite-dimensional normed space. Consider a non-empty convex set $C\subset X$ such that $0\notin C$. Notice that $C$ has a dense and countable subset $\{x_n\}$. $\forall n $ let $C_n= ...
0
votes
0answers
16 views

Gauss-Legendre quadrature error

I'm trying to evaluate the error in Gauss-Legendre quadrature formulae on $[a,b]$. So far I have that the error is less or equal to $$ \frac{f^{(2n)}(\xi)}{(2n)!}\langle p_n,p_n \rangle, \enspace ...