0
votes
1answer
23 views

Show that an operator is weakly compact

If $(X,\Omega,\mu)$ is a finite measure space, $k\in L^\infty(X\times X, \Omega\times \Omega,\mu \times \mu)$ , and $K:L^1(\mu)\to L^1(\mu)$ is defined by $$(Kf)(x)=\int k(x,y) f(y) d\mu(y)$$ show ...
1
vote
0answers
36 views

Weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then $T$ is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B ...
2
votes
1answer
29 views

If any two norms on a vector space are equivalent then the space is finite-dimensional [duplicate]

I need to prove: If any two norms on a vector space are equivalent then the space is finite-dimensional. I am aware of the converse of this result that on a finite dimensional vector space any two ...
0
votes
1answer
16 views

Dual operator of an isometry

If $X,Y$ are Banach spaces and $\phi:X\to Y$ is an isometry, show that $\phi^*$ is surjective. I can use the equality $^\perp(ran \phi^*) = \ker\phi=\{0\}$, and also use the fact that $ran \phi^*$ ...
1
vote
0answers
31 views

Extending linear continuous functions.

Let $E$, $F$ be normed vector spaces and $M$ a subspace of $E$. I'm trying to find an example of a function $f:M\to F$ such that $f$ is linear and continuous but that you can't extend it to ...
0
votes
2answers
33 views

Show that if X is a normed linear space, then any finite-dimensional subspace M of X must be closed.

It suffices to show that any proper subspace M of X is closed, since if M is not proper the result is trivial. I am unsure how to approach this proof. Contradiction seems a little messy, as supposing ...
0
votes
0answers
28 views

Finite dimensional spaces and R^n

I have a couple of questions, any assistance would be appreciated. I know that it can be shown that any finite dimensional space $M$ of dimension $N < \infty$ endowed with an inner product can be ...
2
votes
1answer
47 views

show that every compact operator on Banach space is a norm-limit of finite rank operators (in a particular way, under the given hypotheses)

Let $X$ be a Banach space and suppose there is a net $\{F_i\}$ of finite-rank operators on $X$ such that $(a)$ $\sup_i||F_i||<\infty$ ; $(b)$ $||F_ix-x||\to 0$ for all $x$ in $X$. Show that if $A$ ...
2
votes
0answers
27 views

Maximal chain in the collection of all invariant subspaces for compact operator $K$

Let $X$ be a Banach space over ${\Bbb C}$, and $K\in K(X)$ ($K(X) = $ compact operators space). Show that if ${\cal L}$ is a maximal chain in the $Lat K$ ($Lat K = $ the collection of all invariant ...
2
votes
1answer
51 views

Operators on non-separable Banach spaces have non-trivial invariant subspaces

Show that if $T\in B(X)$ and $X$ is not separable, then $T$ has a nontrivial invariant subspace. I know that $\ker (T)$ and $\operatorname{ran}(T)$ are invariant $T$-subspace. So if $\ker T\neq ...
1
vote
0answers
29 views

functional analysis: show L^1 integral operator has norm 1

I just started my course in functional analysis and have already stumbled across some things I don't understand, which are quite basic :(. In my lecture notes it says: Let $\mu$ be a measure on a ...
2
votes
0answers
23 views

Invariant subspaces and their complements

The following is Exercise 6.4.2 of Conway's Functional Analysis: Suppose $T\in B(X)$ where $X$ is a Banach space. Prove that $M$ is invariant under $T$ if and only if $M^{\perp}$ is invariant under ...
2
votes
1answer
44 views

Compactness of an operator on $c_0$ in terms of its infinite matrix representation

Let $A\in {\cal B}(c_0)$ (${\cal B}(c_0)$ is linear bounded operators on $c_0$) and for $n\geq 1$, define $e_n \in c_0$ by $e_n(m)=\delta_{nm}$. Put $\alpha_{nm}=(Ae_n)(m)$ for $n,m\geq 1$. we have $M ...
1
vote
1answer
34 views

$T\in\mathcal{L}(X,Y)$ maps closed bounded subsets onto closed subsets $\implies$ Range $T$ is closed.

Given two normed spaces $X$ and $Y$ and let $T$ be a bounded linear operator $T:X\to Y$. Assume that $T$ maps bounded and closed subsets of $X$ onto closed subsets of $Y$. Show that the range of $T$ ...
3
votes
1answer
53 views

Give necessary and sufficient conditions for a multiplication on $L^p$ to be compact

Let $(X, \Omega, \mu)$ be a $\sigma-$ finite measure space and for $\phi \in L^\infty(\mu)$ let $M_\phi:L^p(\mu) \to L^p(\mu)$ defined by $M_\phi f = \phi f $ be the multiplication operator. Give ...
0
votes
0answers
35 views

Is every subspace of a normed linear space which is not closed a hyperspace.

Let $B \subset X$ where $X$ is a normed linear space over $\mathbb{R}$ and $B$ is a proper subspace. If $B$ is not closed, is $B$ necessarily a hyperspace(maximal proper subspace) in $X$. I attempted ...
2
votes
1answer
20 views

$\sup$ norm of a function

The following is an example of Murphy's C*-algebras and operator theory: I do not know how he concludes $$\int_0^1 |k(s,t) - k(s',t)||f(t)| dt \leq \sup|k(s,t) - k(s',t)|||f||_\infty$$ Please help ...
3
votes
1answer
52 views

Integral operator on $L^p$ is compact

Let $(X,\Omega,\mu)$ be an arbitrary measure space, $1<p<\infty$ , and $\frac{1}{p}+ \frac{1}{q} = 1$. If $k:X. X\to \Bbb C$ is an $\Omega.\Omega-$ measurable function such that $$M = [\int ...
1
vote
1answer
14 views

Condition under which a locally convex topological vector space becomes a normed linear space

Is this true that a locally convex topological (Hausdorff) vector space becomes a normed space when its local base has only one element, so only one Minkowski functional and so only one seminorm and ...
0
votes
1answer
31 views

Understanding the scalar product in the duality

I was trying to solve an exercise for my class, but then I have found somewhere a solution. I need to understand the meaning of a certain step. The exercise and the solution read as follow. Exercise ...
4
votes
2answers
57 views

Riesz's Lemma for $l^\infty$ and $\alpha = 1$

Riesz's Lemma says the following: Let $X$ be a normed vector space and $Y$ a proper closed subspace of $X$. Pick $\alpha \in (0,1)$. Then $\exists x\in X$ such that $|x|=1$ and $d(x,y) \geq \alpha$ ...
1
vote
1answer
34 views

Proof of compactness for sets of norm equal to one in finite-dimensional normed vector spaces

The proposition I have been trying to prove is that the set $A=\{x\in E:N(x)=1\}$ is a compact subset of the (real) finite-dimensional vector space $E$ for any norm $N:E\to \mathbb{R}$. I am reading ...
6
votes
1answer
147 views

Radon-Riesz & Kadec-Klee

Let us say that a normed vector space has the a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms. b) KK (Kadec-Klee) property ...
1
vote
1answer
48 views

Spectrum of a finite rank operator

If $ T\in B(H)$ is a finite rank operator, then there are orthonormal vectors $e_1,...,e_n$ and vectors $g_1,...,g_n$ such that $Th=\sum_{i=1}^n (h,e_i )g_i$, then we can easily see that $T$ is ...
0
votes
2answers
54 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
2
votes
0answers
28 views

Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
3
votes
0answers
27 views

give necessary and sufficient conditions that every functional in $w^*-cl M$ be the $w^*$- limit of a sequence from M

Let $X$ be a separable Banach space. If $M$ is a linear manifold in $X^*$ give necessary and sufficient conditions that every functional in $w^*-\mathrm{cl} M$ be the $w^*$- limit of a sequence from ...
3
votes
1answer
22 views

convex weak* sequentially closed subset of a separable Banach space implies weak* closed

I'm studying Conway's a course in Functional Analysis by myself. The following is corollary 6.12.7 of this book. If $X$ is a separable Banach space and $A$ is a convex subset of $X^*$ that is weak* ...
2
votes
0answers
34 views

Adjoint of an operator on $C(X)$

Let $X,Y$ be compact and $\tau: Y\to X$ be continuous. If $A:C(X)\to C(Y)$ such that $(Af)(y)=f(\tau(y))$ is a linear and continuous operator, then show that the adjoint operator $A^*:M(Y)\to M(X)$ is ...
0
votes
0answers
56 views

Show that an hyperplane is closed iff f is linear and continuous

I need an help with the following exercise. Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I ...
0
votes
1answer
32 views

Show that an operator is bounded.

Let $\{\alpha_{mn} ;m,n\geq 1\}$ be scalars satisfying a- $M=\sup_n\sum_{m\geq 1}|\alpha_{mn}|<\infty $ , and b- $\sup_n|\alpha_{mn}|<\infty$, then $(Af)(n) = \sum_{m\geq 1}\alpha_{mn} f(m)$ ...
5
votes
1answer
61 views

When is $M+N$ closed

Let $X$ be a Banach space and $M,N$ be closed subspaces. If the range of linear transformation $x\to (x+M)\oplus (x+N)$ from $X$ into $X/M\oplus X/N$ is closed show that $M+N$ is closed. or using ...
0
votes
0answers
16 views

Converse to sequential Banach--Alaoglu [duplicate]

Let $B$ be the closed unit ball of the dual space of a real normed vector space $V$. If $V$ separable then $B$ is sequentially compact in the weak-* topology. What about the converse?
4
votes
1answer
113 views

Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert ...
0
votes
0answers
15 views

Quotient space and continuous linear operator

I'm trying to study some arguments of math by myself and I have some problems to understand the interpretation of the norm about linear operators. The books says that there's a correspondence between ...
1
vote
0answers
50 views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
0
votes
0answers
19 views

Minimization of an evaluation under the weak* topology

I'm self studying (for fun) the book "Functional Analysis, Calculus of Variations and Optimal Control" (by Clarke), and I'd appreciate some feedback for my proposed solution for an exercise from the ...
2
votes
1answer
53 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
0
votes
2answers
54 views

Is my proof correct? Finite-dimensional normed vector spaces

I'm trying to prove that every finite-dimensional normed space is topological isomorphic to $\mathbb{R}^n$. Let $(E,\|\cdot\|_E)$ such that $dimE=n$ and let $$ T:\mathbb{R}^n\to E\\ x\mapsto ...
0
votes
1answer
29 views

Continuity of a map to a Frechet space

Let $(A,\| \cdot \|)$ be a normed space and $B$ be a Frechet space equipped with a family $\{ p_k \}_{k \in \mathbb{N}}$ of seminorms. Let $\phi: A \to B$ be a linear transformation satisfying the ...
2
votes
0answers
32 views

Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
2
votes
1answer
50 views

Implying a positive definite operator

If we are given that $A:V \rightarrow V$ is an operator where $V$ is a real Hilbert space. If we are given that $A$ is bounded, strictly positive $\big(\langle Au,u \rangle > 0$ for all $u \neq ...
2
votes
1answer
81 views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
2
votes
1answer
63 views

How can I prove that $f$ is continuous at $0$?

Let $E$ and $F$ linear normed spaces, and consider a linear function $f:E\to F$ which satisfies for all $(x_n)\in E^{\mathbb{N}}$ such that $x_n\to 0$ then $f(x_n)$ is bounded in $F$. Then I have to ...
4
votes
1answer
54 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
1
vote
1answer
36 views

Exists $l\in X^*$ such that $\|l\|=1, l|_Y=0$ and $l(x_0)=\operatorname{dist}(x_0, Y)$?

Let $(X, \|.\|)$ be a normed space and $Y\subsetneq X$. How can you prove that for $x_0 \in X\setminus Y$ there exists $l\in X^*$ such that $\|l\|=1$, $l|_Y=0$ and $l(x_0)=\operatorname{dist}(x_0, Y)$ ...
1
vote
1answer
40 views

The continuous dual of the reals

I just have a few questions involving the continuous dual of $\mathbb{R}^{N}$. We know that the dual $(\mathbb{R}^{N})^{*}$ of $\mathbb{R}^{N}$ is the space of all linear forms $$a: \mathbb{R}^{N} ...
0
votes
1answer
38 views

Theorems on continuous embedding

Let $X,Y$ be two normed spaces such that $X\hookrightarrow Y$. Show the following: 1) If $A\subset X$ is closed in $Y$ then $A$ is closed in $X$; 2) If $X, Y$ are Banach spaces if $x_n$ tends ...
0
votes
0answers
39 views

Weakly compact closed balls in reflexive space

If it is given that $X$ is a reflexive Banach space. Let $K \subset X$ be a norm closed and norm bounded convex set. I want to show that $K$ is weakly compact. I have the following idea but I am not ...
3
votes
0answers
107 views

Asymptotically isometric copy of $\ell_1$

A Banach space $X$ is said to contain Asymptotically isometric copy of $\ell_1$ if there is a null sequence$(\rho_n)_{n=1}^{\infty}$ in $(0,1)$ and a sequence$(x_n)_{n=1}^{\infty}$ in $X$ so that ...