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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94 views

Is $T$ a compact mapping from $W_{0}^{1,2}\left(\Omega\right)$ into itself? [closed]

Let $\Omega$ be an open bounded subset in $\mathbb{R}^{6}$ and $f$ be in $L^{8}\left(\Omega\right)$. For any $w$ in $W_{0}^{1,2}\left(\Omega\right)$, define $T\left(w\right)$ be in ...
0
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2answers
71 views

Compactness of the identity operator

As far as I know, by Rellich-Kondrachov theorem, we can say $I:H_{0}^{k}\to H_{0}^{m}$, for $m<k$ is a compact operator, where $H_{0}^{k}=\{f\in H_{{}}^{k}|f(0)={f}'(0)=\cdots ={{f}^{(k)}}(0)=0\}$. ...
2
votes
2answers
37 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.
0
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1answer
25 views

Prob. 8, Sec. 4.2 in Kreyszig's functional analysis book: Nonnegativity of a subadditive functional outside a sphere implies nonnegativity

If a subadditive functional $p$ defined on a normed space $X$ is non-negative outside a sphere $\{ \ x \in X \ \colon \ \Vert x \Vert = r \ \}$, then how to show that $p$ is non-negative for all $x ...
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1answer
26 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 ...
4
votes
1answer
66 views

Compute the spectrum of the integral operator $K:L^2([0,1]) \to L^2([0,1])$ defined as $(Kx)(t) = \int_0^t x(s) ds$

Let $K:L^2([0,1])\rightarrow L^2([0,1])$ be the linear operator defined by $$(Kx)(t)=\int_0^tx(s)ds, \quad x \in L^2([0,1]).$$ Now I have to compute the spectrum, but I don't have any idea how to do ...
0
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0answers
20 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 $$ ...
3
votes
1answer
90 views

How to show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete

Show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete. I am not sure where to even start with this question. I'm quite sure it involves Cauchy and also Bolzano-Weierstrass but I'm not sure ...
0
votes
2answers
51 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
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 ...
8
votes
1answer
107 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
1
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1answer
28 views

Prob. 10, Sec. 3.10 in Kreyszig's functional analysis book: Every isometric linear operator on a finite-dimensional inner product space is unitary? [duplicate]

Let $X$ be an inner product space such that $\dim X < \infty$, and let $T \colon X \to X$ be an isometric linear operator. Since $\dim X < \infty$, $X$ is complete and thus a Hilbert space; ...
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2answers
64 views

Distance between two sets

Let $A,B$ be nonempty sets . $ D(A,B)=\inf\{D(a,b) : a\in A , b \in B \} $, let $C=\operatorname{cl}(A) E=\operatorname{cl}(B)$ now how can I prove that : $D(A,B) = D(C,E) $
0
votes
1answer
35 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.
3
votes
2answers
104 views

Bound the norm of the partial trace of an operator on a Hilbert space

Let $H=H_1 \otimes H_2$ a composite Hilbert space and let $A, B$ bounded linear operators on $H$, and we can assume they are trace class. Let $A_2$ we denote the operator on $H_2$ obtained by taking ...
4
votes
2answers
28 views

amenable groups versus amenable graphs

In operator algebras, one is often concerned with amenable groups, defined by one of many equivalent conditions. http://en.wikipedia.org/wiki/Amenable_group#Equivalent_conditions_for_amenability In ...
4
votes
1answer
60 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
28 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 ...
1
vote
1answer
20 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 ...
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 ...
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 ...
0
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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
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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
1
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0answers
40 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 ...
0
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1answer
38 views
+50

Prove or disprove that the following system $\left\{\frac{(-1)^{n-1}}{\pi}\left(\frac{\sin\pi t}{n-1-t}\right)\right\}_2^\infty$ is a Riesz basis.

Prove or disprove that the following system $$\left\{\frac{(-1)^{n-1}}{\pi}\left(\frac{\sin\pi t}{n-1-t}\right)\right\}_2^\infty$$ is a Riesz basis on $L^2(\mathbb R)$. I do not think it is a trivial ...
0
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0answers
22 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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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 ...
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 ...
0
votes
1answer
31 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 ...
0
votes
0answers
23 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 ...
1
vote
1answer
26 views

$L^2(a_1,b_1;H_0^1(a_2,b_2))\subset L^2(a_1,b_1;L^2(a_2,b_2))$ and Convergence

Let $[a_1,b_1]\times[a_2,b_2]\subset\mathbb{R}^2$. Suppose $$u_n\rightharpoonup u\,\,\,\text{ weakly in } L^2(a_1,b_1;L^2(a_2,b_2))$$ and $$\{u_n\}\text{ is bounded in }L^2(a_1,b_1;H_0^1(a_2,b_2)).$$ ...
3
votes
0answers
37 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 ...
0
votes
1answer
77 views

Hamel Basis in Infinite dimensional Banach Space without Baire Category Theorem

Prove that every Hamel basis in an infinite dimensional Banach Space is uncountable without using Baire Category Theory. We are assuming axiom that vector space dimension (if exist) is well-defined. ...
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0answers
37 views

Prob. 9, Sec. 3.10 in Kreyszig's functional analysis book: The image of ann isometric non-unitary operator on a Hilbert space

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a linear operator such that $T$ is isometric but not unitary. Then how to show that the image $T[H]$ is a proper closed subspace of $H$? My ...
10
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2answers
154 views

Can you equip every vector space with a Hilbert space structure?

Suppose that we have a vector space $X$ over the field $\mathbb F \in \{ \mathbb R, \mathbb C \}$. Question: Does there exist a Hilbert space $\widehat X$ over $\mathbb F$ such that $\widehat X$, ...
2
votes
1answer
49 views

Can every vector space (over $\mathbb{R}$ or $\mathbb{C}$) can be a Banach space (or Hilbert space)?

For a vector space $V$ over $\mathbb{R}$ (or $\mathbb{C}$) with Hamel basis of cardinality $\kappa$ such that $\kappa^{\aleph_0} = \kappa$, can we define inner product(or norm) on $V$ such that $V$ is ...
0
votes
1answer
40 views

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$ where $x=(x_1, x_2, x_3) \in \mathbb{R}^3$ I was trying to look at Hessian matrix and use Sylwester theorem, but I see that ...
2
votes
1answer
49 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 ...
2
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0answers
36 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 ...
0
votes
1answer
62 views

Reducing Spaces: Domain

Problem Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Denote for readability: $$\mathcal{D}:=\mathcal{D}(N)=\mathcal{D}(N^*)$$ ...
1
vote
2answers
73 views

proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that ...
1
vote
1answer
38 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.
1
vote
1answer
47 views

Borel Measures: Lusin

I'm trying to self-learn. Given the complex plane $\mathbb{C}$. Consider a Borel measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\mu\geq0$$ Regard a measurable: ...
2
votes
1answer
48 views

Reference about Sobolev spaces

I'm looking some book from which I could learn more about Sobolev spaces. I'm interested rather in abstract theory: some topics which I would like understand in detail include: general construction ...
5
votes
1answer
65 views

Equivalent formulations: pure contraction

I want to prove the following equivalence: let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. TFAE: $\|Tx\|<\|x\|$ for each $x\in H\setminus\{0\}$ $\|T\|\leq1$ and ...
1
vote
1answer
40 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 ...
2
votes
0answers
25 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
1
vote
1answer
29 views

If $S:(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$ Why does $0 \in \sigma (S)$? and what is $\dim(R(S))$ in $\ell^2$?

Consider the unilateral shift $S$ on $\ell^2$ defined by $$(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$$ Why dose $0$ is eigenvalue of $S$? and what is $\dim(R(S))$ in $\ell^2$?( Range of $S$)