A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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12
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0answers
133 views

Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? ...
2
votes
2answers
62 views

Cauchy Sequence in Normed Space

Let $(E, ||\cdot ||)$ be a normed space and let $(x_n)$ be a sequence in $E$. Show that the following conditions are equivalent: (a) $(x_n)$ is a Cauchy sequence. (b) For every increasing function ...
0
votes
3answers
60 views

The set $A=\{ x :\|x\|=1 \}\subset \mathbb{R}^n$ is compact in $\mathbb{R}^n$ under euclidean norm.

The set $A=\{ x \mid \|x\|=1 \}\subset \mathbb{R}^n$ is compact in $\mathbb{R}^n$ under Euclidean norm. If I can show this is closed and bounded then we are done. It seems bounded but I am not sure ...
0
votes
1answer
24 views

Let $A=\{ f \in \ell^1\, :\,\text{ for each natural number}\, n\,\text{we have}\, |f(n)|<1/2^n \}$. find the closure of $A$ in $\ell^1$.

Let $A=\{ f \in \ell^1\, :\,\text{ for each natural number}\, n\,\text{we have}\, |f(n)|<1/2^n \}$. find the closure of $A$ in $\ell^1$. I know the interior of A is empty. for the case of getting ...
1
vote
1answer
24 views

A is totally bounded in $l_1 $ if and only if $ (e_n ) \in l_1 .$

Let $(e_n)$ be a sequence of positive real number. Show that A ={$ x=(x_n) \in l_1: |x_n| \le e_n (n \in N ) $ is totally bounded in $l_1 $ if and only if $ (e_n ) \in l_1 .$ I have seen similar ...
2
votes
0answers
55 views

Name for the universal normed space associated to a seminormed space

If $(V,p)$ is a seminormed space, then $(V/N,\overline{p})$ is a normed space, where $N=\{x \in V : p(x)=0\}$ and $\overline{p}(x \bmod N) = p(x)$. My question is as follows: Is there a common name ...
1
vote
3answers
383 views

Norms and equivalence classes question

Let $f\in C[0,1]$. Recall two of the norms we considered in class: $$\|f\|_\infty = \sup_{t\in[0,1]}|f(t)|, \quad \|f\|_1 = \int_0^1|f(t)|\ \mathsf dt. $$ Consider the space $C^1([0,1])$ of ...
0
votes
1answer
22 views

Sequence space norm on $\mathbb{R}^2$

I am working on an assignment so I won't be stating the actual question, but it's about showing extensions of a linear functional in $\mathbb{R}^2$ and it says, I quote Determine all it's linear ...
1
vote
2answers
33 views

Prove that any projection on a normed linear on a subspace satisfies $\|I-P\|\geq 1$

Let $M$ be a subspace of a normed linear space $N$. Let $P$ be a (continous) projection on $M$. Then $$\|I-P\|\geq 1 $$
1
vote
0answers
48 views

Hahn-Banach and hyperplane separation

Let's take a look at the following result, based on Hahn-Banach (extension of bounded linear (real-valued) functionals): Let $X$ be a normed space, $U$ a subspace of $X$ and $u_0 \in X$ with ...
0
votes
3answers
31 views

Normed Linear Space ,$p \neq 2$ is $\left \| f\right \|_{p}= \sqrt{f,f}$ for each $ f \in L^P([0,1])$?

For $p \neq 2$, is there an inner product $< ., .>$ on $L^P([0,1])$ such that $\left \| f\right \|_{p}= \sqrt{f,f}$ for each $ f \in L^P([0,1])$? is it true?for P=2 norm is induced by inner ...
1
vote
1answer
34 views

Operator norm: Show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\| \leq 1 , f \in Y^*, \|f\| \leq 1 \}$

Good day, As stated in the title, I have to show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\|_X \leq 1 , f \in Y^*, \|f\| \leq 1 \}$ where $\| \cdot \|$ is the operator norm, i.e. for $X,Y$ vector ...
2
votes
1answer
84 views

$\forall$ $\epsilon > 0$, $\exists \ \delta > 0$, $\|s\|, \|t\| < \delta \implies$ $|f(x_0 + s) - f(x_0 + t) - Df(x_0)(s-t)| < \epsilon \|s - t \|$

$E$ is a normed vector space and $\Omega \subset E$ is open. Let $f: \Omega \to \Bbb R$ be Fréchet differentiable on $\Omega$. Let $Df$ be the derivative map of $f$; $Df: \Omega \to \mathcal L(E, ...
3
votes
1answer
108 views

Every Banach space is isomorphic to $\ell_1/A$ for some closed $A\subset \ell_1$

How to prove the following mind-blowing fact? Let $(X, \|\ \|)$ be a separable Banach space, $\ell_1\subset \mathbb{R}^\infty$ - the space of absolutely summable scalar sequences. Then there ...
2
votes
0answers
37 views

Test of a normed vector space to be a direct sum with closed summand

Let $X$ be a normed vector space, $X_1$ subspace of $X$. There exists such closed subspace $X_2$ of $X$ that $X = X_1 \oplus X_2$ if one of the following conditions stands: $\dim X_1 < \infty$; ...
0
votes
1answer
25 views

Exists $y \in K$ such that $\|y-x_0\|=\inf\{\|x-x_0\| :x \in K\}$ for disjoint closed sets $K$ and the set of $\{x_0:\|x_0\|≥b\}$

$K$ closed, $x_0 \in R^n$. $K \subset B(0,a)$ and $\|x_0\|≥b>a$ I'm looking for the idea behind displaying the existence of $y \in K$ such that: $$\|y-x_0\|=\inf\{\|x-x_0\| :x \in K\}$$ which ...
2
votes
1answer
58 views

The sum of subspace of finite dimension with a closed set is closed

In Banach-Hilbert Spaces, Vector Measures and Group Representations, P142 7-4.9 Corollary Let $M$ be a closed vector subspace of a normed space $E$. Then for every finite dimensional vector subspace ...
0
votes
1answer
47 views

Show f: l^2 ->l^1, defined by f({x_n}) = {x_n/n}, is uniformly continuous

Let $f$ be a function defined on $\ell^{2}$ to $\ell^1$ by $f(x) =(x_n/ n)$. Show $f$ is uniformly continuous. My attempt at the proof: Let $\{a_n\}$ and $\{b_n\}$ be sequences in $\ell^2$ Let ...
3
votes
1answer
74 views

Sum of open and closed sets

Let $A,B$ subsets of a normed space $(X,\|\cdot\|)$ and $A+B=\{a+b\mid a\in A,\, b\in B\}$ I need help with the next proofs, I can't figure how to begin the proofs: (a) If $A,B$ open then $A+B$ open ...
0
votes
1answer
11 views

Bounded vectors give bounded scalars, finite-dimensional vector space.

Assume that we are in a finite dimensional vector space, with basis $\{v_1,v_2,\ldots,v_n\}$. Assume also that we have a sequnce of bounded vectors, $\{x_i\}$, that is $\|x_i\|<M$ for some real ...
2
votes
1answer
30 views

Hahn-Banach separation theorem with a countable subset of functionals

For a separable Banach space $X$, the unit sphere of $X^*$ always contains a countable set $D$ such that $$ \left\Vert x \right\Vert = \sup_{f \in D} \left\vert f(x) \right\vert \qquad \mbox{ for ...
0
votes
1answer
17 views

Normed-Space; lower bound needed for ||x||+||y||−||x+y||

Looking for an $f(\cdot)$ such that $||x|| + ||y|| - ||x+y|| \geq f(||y-x||)$ Here we had a question for the lower-bound version, where the result is that $||x|| + ||y|| - ||x+y|| \leq ||y-x||$
2
votes
1answer
44 views

Normed-Space; bound needed for $||x|| + ||y|| - ||x+y||$

Given x and y, is there any way we can express $||x|| + ||y|| - ||x+y||$ in terms of $||y-x||$? Even a bound where $||x|| + ||y|| - ||x+y|| \leq f(||y-x||)$ for some $f(\cdot)$ would be desirable. ...
2
votes
0answers
26 views

Operator norm of matrix of scalars regarded as matrix with entries in the unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+=\{(a,z):a\in A,z\in\mathbb{C}\}$ with product $(a,z)(b,w)=(ab+zb+wa,za)$ and norm $||(a,z)||=||a||+|z|$. Equip $M_n(A^+)$ with the operator norm by ...
0
votes
1answer
34 views

Show that C^α([0, 1]) is of first category in C([0, 1]).

Recall that for any $0 < α < 1,$ the space $C^\alpha ([0, 1])$ is the set of continuous functions on $[0, 1]$ with norm of f = sup |f| + $ sup \ x\ne y$ |f(x) − f(y)|/|x − y|^α< ∞, equipped ...
0
votes
1answer
16 views

The set A is in a normed vector space W. $S=\bar A \cap \bar {A^c}$.

The set A is in a normed vector space W. $S=\bar A \cap \bar {A^c}$ (S is the intersection of the closure of A and the closure of $A^c$). Is there a set A in W=R for which S=the set of rational ...
1
vote
1answer
24 views

Characterization of square-summable sequences

I'm curios whether or not the following implication is true: If $x_{n} \notin \ell^2{(\mathbb{N})}$, is there necessarily a sequence $y_{n} \in \ell^{2}(\mathbb{N})$ such that $x_{n}y_{n} \notin ...
0
votes
0answers
31 views

Can we show that $(E\times \mathbb R)^*=E^* \times \mathbb R$ where $E$ is a Banach space?

Can we show $(E \oplus \mathbb R)^* \cong E^* \oplus \mathbb R$, where $E$ is a Banach space and $E^*$ is the dual space of $E$? What if $E$ is just a normed space or even a topological space? To be ...
0
votes
0answers
20 views

Norm of matrix in $M_2(\mathbb{C})$ as operator on $\ell_p^2$

Let $\ell_p^2$ denote $\mathbb{C}^2$ with the $\ell_p$ norm where $p\in[1,\infty)$. I did a computation that indicates that if $A$ is a matrix in $M_2(\mathbb{C})$, then the norm of $A$ as an operator ...
0
votes
1answer
26 views

Convex subsets and Linear functionals

Let $E$ be a convex subset of a normed space $X$ and $x\in E$. Then $x\in \overline{E}$ if and only if $\Re f(x)\geq 1$ for every $f\in X'$ such that $\Re f\geq 1$ on $E$ and $\Re f(x)\leq 1$ for ...
0
votes
1answer
28 views

Give an example of a linear mapping from a normed space into a normed space which is not continuous. [closed]

Give an example of a linear mapping from a normed space into a normed space which is not continuous. I can't think of anything. Any help would be very appreciated.
1
vote
1answer
34 views

Prove that a linear mapping from a normed space into a normed space is continuous if and only if it maps bounded sets to bounded sets.

Prove that a linear mapping from a normed space into a normed space is continuous if and only if it maps bounded sets to bounded sets. I have an idea if the sets were Cauchy, but I can't assume that ...
2
votes
0answers
27 views

function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$ [duplicate]

I'm searching for a function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$. And it has to be $|\Omega |<\infty$. I tried $f(x)=\frac{1}{2\sqrt{x}}$ and $\Omega= ...
2
votes
1answer
34 views

Distance of $x$ to kernel of bounded linear functional is the norm of the functional at $x$?

Let $X$ be a Banach space and let $f\in X^*$ have norm $1$. Prove that $x\in X\implies d(x,\text{ker} f)=|f(x)|$. I have managed to prove that $d(x,\text{ker}f) \geq |f(x)|$, using a theorem that ...
1
vote
1answer
47 views

Application of Hahn Banach Separation theorem

I am solving an exercise (not Homework).. Let $E_1$ and $E_2$ be non empty disjoint convex subsets of $X$, with $E_1$ compact and $E_2$ closed in $X$. Then there are $f\in X'$ and $t_1,t_2$ in ...
0
votes
1answer
55 views

Open Ball and Lipschitz Equivalence equivalence

I am trying to show that two norms $\|\cdot\|$ and $\|\cdot\|^\prime$ are Lipschitz equivalent if and only if there exist numbers $r,R >0$ such that $B_r \subseteq B_1^\prime \subseteq B_R$ where ...
3
votes
0answers
60 views

completeness of $l_1 ^\infty$

I'm trying to prove that $l_p ^\infty $ is complete for each $p\geq 1$ but only with the definition of $\varepsilon$-$N$. I know that this have been proved in other posts here but I couldn't find a ...
1
vote
1answer
30 views

Definition of space $L_f^2$ where $f$ is a function?

http://it.tinypic.com/r/2iqjvbl/9 Hi guys! I'm writing my thesis for my degree and it's about Sturm-Liouville theory applications. I'm using the book "Al-Gwaiz M.A. Sturm-Liouville Theory and Its ...
1
vote
1answer
27 views

Convergent sequence on unit sphere

Suppose $x_n$ is a bounded sequence in a vector space $V$ with norm $||\cdot||$. Show that if: $$\hat{x}_n=\frac{x_n}{||x_n||}\;\;\text{converges}\Rightarrow x_n\;\text{has a convergent ...
0
votes
1answer
28 views

Geometry of a Cauchy sequence in a normed space

A sequence in a normed space $X$ is called a Cauchy sequence if and only if for every $\epsilon > 0$ there exists an integer $N\in \Bbb N$, such that $\|x_n-x_m\|\lt \epsilon$ for all ...
1
vote
1answer
13 views

Deducing equivalence between norms from simple condition

Let $||\cdot||,\;||\cdot||'$ be norms on $V$. Suppose for some $a,b>0$ we have: $$||x||<a\Rightarrow ||x||'<1\Rightarrow ||x||<b$$ Show that $||\cdot||,\;||\cdot||'$ are Lipschitz ...
2
votes
1answer
35 views

Finding closure/interior of subset of function space

Consider the subset $$A=\left\{f\in C(\Bbb R): |f(x)|< \frac{1}{1+|x|} \, \text{for all } x\in \Bbb R\right\}\subset \left\{f\in C(\Bbb R): \lim_{|x|\to \infty}f(x)=0 \right\}=X.$$ where $X$ is ...
0
votes
1answer
26 views

Do I Understand Closed Versus Complete in Metric, Normed and Inner Product Spaces?

I've looked at a number of references on this including some questions on stack exchange. Am I correct if I summarize by stating the following ? (1) A space C (metric, normed, or inner product) is ...
2
votes
1answer
48 views

Normed Linear Space - maximum norm vs. $||f||_1$

For $f$ in $C[a,b]$ define $$|| f ||_1 =\int_a^b |f|.$$ a. Show that this is a norm on $C[a,b]$. b. Show that there is no number $c \geq0$ for which $$||f||_{max} \leq c ||f||_1 \ for \ all \ f \ ...
1
vote
2answers
48 views

Define $g :\ell_2 \to \mathbb R$ by $g(x)= \sum_{n=1}^{\infty} \frac{x_n}n$. Is $g$ continuous?

Define $g :\ell_2 \to \mathbb R$ by $$g(x)= \sum_{n=1}^{\infty} \frac{x_n}n $$ Is $g$ continuous? I need to solve this but I could not see how to tackle it? any hints or suggestion?
4
votes
0answers
91 views

Are uncountable “Schauder-like” bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
2
votes
1answer
28 views

Norms on unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+$ be its unitization. Then $||(a,z)||_1=||a||+|z|$ is a Banach algebra norm on $A^+$. Can we also make $A^+$ a Banach algebra by giving it the norm ...
1
vote
1answer
48 views

Show that $\exists$ an inner product in $X$ such that $<x,x> =||x||^2$ for all $x \in X$

Problem: Let $X$ be normed space. If on every two dimensional subspace $Y$ of $X$, there is an inner product $<,>_Y$ such that $<y,y>_Y=||y||^2$ for all $y\in Y$. Then there is an inner ...
4
votes
1answer
57 views

Confused by peculiar norm

Let $X$ be an infinite subset of $ [0,1]$. In an exercise I am considering the norm on $P([0,1])$ (polynomials on unit interval) defined by: $$||p||_X=\sup_X |p|$$ My question is, how do I make sense ...
0
votes
1answer
158 views

Uncountably many norms such that no two are Lipschitz equivalent

I am struggling with the following question: Is it possible to find uncountably many norms on $C[0,1]$ such that no two are Lipschitz equivalent? I had thought about trying to define norms for each ...