2
votes
2answers
148 views

Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
2
votes
1answer
44 views

Why are “not bounded” operators not everywhere defined?

Let $X, Y$ be Banach spaces, $\mathcal{D}(T)$ a subspace of $X$, and $T\colon X\to Y$ a linear map. Such a $T$ is commonly called an unbounded linear operator, where unbounded just means that the ...
1
vote
1answer
52 views

Bochner: Lebesgue Obsolete?

Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion ...
0
votes
1answer
30 views

Understanding the proof that $c_0$ is a closed subspace of $\ell^\infty$

The problem is given: source Let $c_0$ be a space of real sequences $x = \{x_n\}_{n = 1}^\infty=0$ converging to $0$. Let $\ell^\infty$ be a set of real sequences $w = \{w_k\}^\infty_{k=0}$ ...
3
votes
1answer
40 views

tough lp inequalities

Let $1<p<\infty$. If possible, find a positive decreasing sequence $w_1>w_2>\cdots$ such that $\lim w_i=0$, and a (uniform) constant $K>0$, such that ...
1
vote
0answers
14 views

Direct sum decomposition of vector spaces and their tensor powers

Let $V$ be a locally convex vector space and let $U$ be a finite-dimensional subspace of $V$. The Hahn-Banach theorem guarantees that there exists a closed subspace $W$ of $V$ such that $$V=U\oplus ...
1
vote
1answer
19 views

Quotient of a Banach space X gets quotient topology under standard norm induced from X

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. Then there is a canonical norm on $X/Y$. I want to show that this norm induces quotient topology on $X/Y$. Any hint/solution? I was ...
0
votes
1answer
25 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
38 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 ...
1
vote
0answers
45 views

Continuity of a linear map on Banach space.

A function $f$ on $X$ is said to separate points of $X$ if for $x_1\neq x_2$ then $f(x_1)\neq f(x_2)$. Suppose $A$ is a subset of $Y^{*}$, the space of functionals on a Banach space $Y$, which ...
2
votes
1answer
42 views

Dimension for a closed subspace of $C[0,1]$.

Let $X \subset C^1[0,1]$ be a closed subspace of $C[0,1]$ (with sup norm). Prove that $X$ has to be finite-dimensional.
1
vote
3answers
51 views

Basis of eigenfunctions in Banach space

I have a question about proving the existence of a basis of eigenfunctions. Assume we have a compact operator $L:\mathcal{B}\rightarrow\mathcal{B}$, where $\mathcal{B}$ is a Banach space of analytic ...
2
votes
2answers
38 views

Definition of continuous spectrum of a bounded operator

Let $T$ be a bounded operator acting on a Banach space $X$. The point spectrum $\sigma_p(T)$ is of $T$ is defined to be $$\sigma_p(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has nonempty kernel}\}$$ ...
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 $ran \phi^*$ ...
2
votes
1answer
30 views

The image of Banach space under its embedding provided by the Banach-Mazur theorem

It is a very nice argument of Banach and Mazur which they use to show that every Banach space $X$ is isometric to a subspace of the space $C(B_{X^*})$, where $B_{X^*}$ is the unit ball of the dual ...
3
votes
1answer
61 views

Relative countable weak$^{\ast}$ compactness and sequences

I am finding serious difficulties in understanding some things about relative countable compactness and the use of sequences in proving it by my functional analysis text, Kolmogorov-Fomin's. For ...
2
votes
1answer
49 views

$A^{\ast}$ compact $\Rightarrow A$ compact

I read that if $A:E\to E$ is a bounded linear operator where $E$ is a Banach space and $A^{\ast}$ is a compact operator, then $A$ is a compact operator. I know that the converse is true (th. 4 here), ...
1
vote
1answer
19 views

Compactness of the image of $S^{\ast}$ throw compact operator

I read, on an Italian language version of Kolmogorov-Fomin's Introductory Real Analysis, that, for any Banach space $E$, the unit closed sphere $S^{\ast}$ of $E^{\ast}$ is compact in the $\ast$-weak ...
1
vote
1answer
35 views

Non-separable Hilbert spaces in duals

A topological space $X$ satisfies the countable chain condition if every family of pairwise disjoint open sets in $X$ is countable. I am looking for a reference to the following fact: Suppose that ...
1
vote
1answer
41 views

Containment of $c_0$ or $\ell_p$

Suppose that $(x_n)$ is a sequence of unit vectors in a Banach space $X$ such that $$\mbox{dist}(x_m, S_{X_n})=1$$ for all $m > n$. Here $S_{X_n}$ stands for the unit sphere $\mbox{span}\{x_1, ...
0
votes
1answer
22 views

Bounded resolvent

I am finding a somewhat obscure wording on my functional analysis text. $A$ is a linear operator on a complex Banach space $E$, which I'm almost sure to mean that $A$ is $E\to E$. Then I read that if ...
0
votes
1answer
20 views

Banach Algebras: Continuity of Inversion?

Context: This question is related to this thread: Spaces of Functions Given a topological space $X$ and a Banach algebra with unit $B$. Consider a continuous map $F:X\to B$ that is invertible ...
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
43 views

Reference request for the proof of the Brodskii–Milman fixed point theroem for isometries

Can any one help me to access the paper M.S Brodskii and D.P Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948) 837–840 in Russian? or to prove the theorem If $K$ is 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
1answer
48 views

The space $ \left( \sum \ell_p^n \right)_2$ is reflexive.

Let $\ell_p^n:= (\mathbb R^n, \|\cdot\|_p)$. I want to show that the space $$ \left(\sum_{n=1}^\infty\ell_p^n \right)_2 := \left(\left\{ (x_n)_{n \in \mathbb N} : x_n \in \ell_p^n ...
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 ...
4
votes
0answers
27 views

Differentiation in Besov-Zigmund spaces

This is my second question in a short time on Besov spaces. I apologize. I am having a rough time with them and I really need to understand this spaces quickly. The besov spaces ...
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
52 views

Proving $u$ is compact whenever $u^\ast$ is

Let $X,Y$ be Banach spaces and let $u: X \to Y$ be a linear operator. Let $u^\ast: Y^\ast \to X^\ast$ denote its transpose and assume that $u^\ast$ is compact. I am trying to prove that $u$ is ...
5
votes
1answer
69 views

Is $C^{\infty}([0,1])$ a Banach space?

I have read that the answer is no, but I am unable to prove it. Give $C^{\infty}([0,1])$ the metric $$d(f,g) = \sum_{j=0}^{\infty} 2^{-j} \frac{||(f-g)^{(j)}||}{1 + ||(f-g)^{(j)}||}$$ associated to ...
0
votes
2answers
49 views

Extending Positive Functionals: Linearity

How does regularity provide linearity? Given the full Banach space of bounded functions over a suitable set: $$\mathcal{B}:=\{f:\Omega\to \mathbb{C}:\|f\|_\Omega<\infty\}$$ and a linear subspace ...
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 ...
2
votes
1answer
22 views

An abstract a priori estimate in finite element method

Let $V$ and $K$ be Banach spaces (with norms $\|\cdot\|_V$ and $\|\cdot\|_K$ resp.) and suppose that there is a compact linear embedding $K\hookrightarrow V$. Furthermore, let $P_n$ be a family of ...
0
votes
1answer
25 views

Extending Banach Space of Functions

The idea is that one could in principle consider the space of functions: $$\{f:\Omega\to V\}$$ with pointwise operations and uniform convergence: $$f_\lambda\to f:\iff\|f_\lambda-f\|_\infty\to 0$$ ...
2
votes
1answer
41 views

Three-space property

I have found two definitions of a three-space property. One definition is: $(P)$ is a three-space property if whenever $E$ Banach space, $F\subseteq E$ is a closed linear subspace and two of the ...
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 ...
2
votes
0answers
46 views

Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
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 ...
0
votes
2answers
52 views

Space of Functions: Characterizations of Positivity

Context The problem here is about the characterization of positivity for real or complex valued functions: $$\sigma(f)\geq 0\iff\sigma(f(x))\geq 0\text{ for all }x\in X\iff f(x)\geq 0\text{ for all ...
3
votes
2answers
32 views

Banach Spaces: Totally Bounded Subsets

As an easy consequence of Riesz' lemma it is known that infinite dimensional Banach spaces possess bounded subsets that fail to be totally bounded. On the other hand in finite dimensional Banach ...
3
votes
1answer
53 views

Sums of special vectors

Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary ...
0
votes
1answer
56 views

Banach Spaces: Totally Bounded vs. Bounded

Are the finite dimensional Banach spaces precisely those ones in which subsets are totally bounded iff they're bounded?
10
votes
1answer
90 views

When is an operator on $\ell_1$ the dual of an operator on $c_0$?

Suppose $T:\ell_1\to\ell_1$ is a continuous linear operator. When can we say that $T$ is a dual, or adjoint, of an operator on $c_0$? In other words, under what conditions can we find a continuous ...
1
vote
0answers
32 views

Bounded below adjoint operators

Let $T\colon X\to Y$ be a bounded linear operator. Suppose that $Z$ is a subspace of $Y^*$ such that $T^*$ is [bounded below][1] on $Z$. Denote by $\text{w*-dens}\, Z$ the minimal cardinality of a ...
2
votes
1answer
55 views

Discontinuous seminorm on Banach space

We have known that if $X$ is a Banach space and $\sum_{n=0}^{\infty}x_n$ is an absolutely convergent series in $X$ then $\sum_{n=0}^{\infty}x_n$ is a convergent series. Moreover, we have $$ (*)\quad ...
5
votes
0answers
63 views

Weak*-complemented subspaces of $\ell_\infty$

Consider $\ell_\infty$ as $\ell_1^*$. Let $X$ be an infinite-dimensional complemented subspace of $\ell_\infty$ (in partiuclar, $X$ is isomorphic to $\ell_\infty$). Can we find a further subspace ...
3
votes
1answer
59 views

Small $\ell^p$ spaces are obtainable from $L^p$

I've seen that in a lot of books there is written that $$l^p=L^p(X,\Sigma,\mu),$$ where $X=\Bbb N, \Sigma=P(\Bbb N), \mu=\#$, ($\#$ is the counting measure). I would like to see how to prove it, ...
2
votes
0answers
25 views

Does a strictly convex and weak metrizable unit sphere of a Banach space imply locally uniform convexity?

I'm trying to find a proof for this question Let $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology. Does $X$ admits ...