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
31 views

$(X/Y)^*$is isometrically isomorphic to $Y^⊥$

Let X be a Banach space with a closed subspace Y, We define the dual mapping $ \pi^*:(X/Y)^* → X^*$ by $\pi^* (\beta)=\beta\circ\pi$ then $(X/Y)^*$is isometrically isomorphic to $$Y^\perp:=\{f∈ ...
1
vote
1answer
40 views

Comparing two linear functions

Let $X$ be a Banach space and let $h:X\to\Bbb C$ and $f:X\to\Bbb C$ be two bounded linear functions such that if for some $x\in X$ we have $f(x)=0$ then $h(x)=0$. Prove that there exists a ...
0
votes
1answer
36 views

existence of an “inverse” adjoint in Banach Spaces

Let X,Y be Banach spaces and S a bounded operator $S: Y' \rightarrow X'$ where $'$ denotes the dual space or the adjoint operator depending on what it is on. Then $$ \exists \ \ T \in B(X,Y) : T'=S ...
3
votes
1answer
33 views

Does completing a normed space commute with taking quotients?

Let $X$ be a normed vector space and $Y \subset X$ a closed subspace. We consider the quotient $X / Y$ and equip it with the quotient norm. Then we may form the completion $\overline{X / Y}$. We ...
2
votes
1answer
51 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
4
votes
1answer
74 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
-1
votes
1answer
27 views

Finding a condition to make a map a contracting map

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
2
votes
0answers
188 views

Hahn-Banach via Hamel Basis

my question for tonight: Is there a proof for Hahn-Banach using a Hamel Basis? I know, the proof for existence of Hamel Bases uses already Axiom of Choice, but I'd like to apply this without refering ...
3
votes
1answer
61 views

Relationship between a finite codimensional subspace of dual space and the annihilator

Notation: $X$ is a banach space, $X'$ is the dual space to $X$. When $V \subset X'$, we write $\ker V = \cap_{l \in V} \ker l$, and when $W \subset X$, we write $ann \; W = \{l \in X' \mid l(w) = 0 ...
2
votes
1answer
59 views

What is the dual of $(\mathbb{R}^n,∥\cdot∥_{\infty})$

I know that the dual of $(\mathbb{R}^n,∥\cdot∥_{p})$ is $(\mathbb{R}^n,∥\cdot∥_{q})$ with $\frac{1}{p}+\frac{1}{q}=1$ But does this also hold when $p=\infty$ and if so what is the proof?
0
votes
0answers
47 views

Dual Spaces vs subspaces

This seems simple, but I just can't quite convince myself. I'm sure someone out there can help. Let U be a finite dimensional vector space with dual U$^*$ and let B = {$f_1,f_2,...f_n$} be a basis ...
1
vote
1answer
118 views

How to show that the dual of $(\mathbb{R}^n,\|{\cdot}\|_p)$ is $(\mathbb{R}^n,\|{\cdot}\|_q)$?

I am trying to brush up on my functional analysis and I learn some $L_p$ spaces since I was never formally intrduced to them through courses. I wanted to know if anyone could offer me a proof or give ...
3
votes
1answer
169 views

Closed-form expressions for dual norms of real normed vector spaces

Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\|v\|$. Then say that $V^*$ is the dual space of linear functionals on $V$. The "dual ...
0
votes
1answer
91 views

Invariant subspace associated with a complex eigenvalue

Let $E$ be a Banach space over the reals and $T:E\to E$ a linear operator. Suppose that $\mu=re^{i\theta}$ is a eigenvalue of $T$ with $r>0$. How can one find two vector $z_1,z_2\in E$ such that if ...
2
votes
1answer
51 views

If $(M_{\lambda})$ be a chain of closed affine subspaces of $X$ $\Longrightarrow$ $\displaystyle \bigcap_{\lambda\in L} M_{\lambda}\neq \emptyset \;$?

Let $X$ be a Banach space Let $(M_{\lambda})_{\lambda \in L}$ be a chain of closed affine subspaces of $X$ We can say that $$\displaystyle \bigcap_{\lambda\in L} M_{\lambda}\neq \emptyset \;\;?$$ ...
3
votes
3answers
117 views

How to prove that sequence spaces $l^{p}, l^{\infty}$ and function space $C[a. b]$ are of infinite dimension

I am studying about the sequence space $l^{p}, l^{\infty}$ and function space $C[a. b]$. It is mentioned in the book that all of these spaces are of infinite dimension. I want to prove that these ...
1
vote
0answers
54 views

Definition of a countable direct sum of subspaces of a Banach space

Let $X$ be a separable Banach space and $K\subseteq X$ a subspace. Let $\{H_i\}_{i\in I}$ be a countable collection of subspaces of $X$. Is it correct that $K=\bigoplus H_i$ iff every element $k\in K$ ...
0
votes
1answer
121 views

Image of a set under a mapping

I need to show that the image of the closed unit ball in $\mathbb{C}$, under the polynomial mapping $p(x) = (1-x)^2$ is the cardioid: ${re^{i\theta} : 0 \leq \theta < 2π, 0 ≤ r ≤ 2 + 2 ...
1
vote
0answers
76 views

Existence of a transpose of a linear operator

Let $X,Y$ be two linear spaces and let $b:X\times Y\to\Bbb R$ be a bilinear map. For any linear operator $A:X\to X$ we can define its $b$-transpose acting on $Y$ by the system of the following ...
1
vote
1answer
59 views

Finite dimensional quotient $\Rightarrow$ closedness?

Let $X$ be a Banach space and $V\subset X$ a subspace. Suppose that the quotient space $X/V$ is finite dimensional. Is $V$ then closed? In other words: if $V$ is not closed, then can $X/V$ be finite ...
1
vote
1answer
153 views

An upper bound for $\|(\lambda-A)^{-1}\|$?

Let $A$ be a k-by-k matrix and $\sigma(A)$ its spectrum, or the collection of eigenvalues of $A$. If we know $\lambda\notin\sigma(A)$, then $\lambda$ is at a positive distance to all points in the ...
3
votes
1answer
318 views

What is the precise definition of predual

How does one define "predual" and the surrounding notions? More specifically: Why must there be only one predual of $X$ when $X$ is a Banach space? What is the correct notion of similarity here ...
1
vote
1answer
126 views

Immediate predecessor in a chain of subspaces

Let $\mathcal{C}$ be a chain of subspaces of a Banach space $\mathcal{X}$. For each $\mathcal{Y}\in\mathcal{C}$, define its immediate predecessor ...
2
votes
1answer
86 views

Almost invariant subspaces for WOT closure of an algebra of operators

Let $X$ be a Banach space and $\mathcal{C}\subset\mathcal{L}(X)$ be a collection of bounded linear operators. A subspace $Y$ is said to be almost invariant under $\mathcal{C}$ if for each ...
1
vote
1answer
113 views

tensorisation of linear map

Let $X$ be a Banach space and $T \colon \ell^2\rightarrow \ell^2$ be a bounded linear map. Suppose that the linear map $T\otimes Id_ {X}:\ell^2\otimes X\rightarrow \ell^2\otimes X$ which maps $e_i ...
3
votes
2answers
471 views

projection operators on topological vector spaces

Suppose $A \in \mathbb{R}^{m\times n}$. Then there exists a projection matrix $P$ onto the range of $A$. In other words, there exists a matrix $P \in \mathbb{R}^{m\times m}$ such that $P^2=P$, and ...