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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2
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1answer
85 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
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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
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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
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1answer
67 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 ...
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1answer
48 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
82 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
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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
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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
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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
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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
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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
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1answer
26 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
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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
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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
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1answer
28 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
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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.
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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
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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
40 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
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1answer
50 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
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1answer
58 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
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0answers
63 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
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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 ...
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1answer
29 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
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1answer
29 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
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1answer
14 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
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1answer
41 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
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1answer
27 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
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1answer
49 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
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0answers
95 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
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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
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1answer
50 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
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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
173 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 ...
0
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0answers
30 views

Proving that given this property then the norm is induced by a inner product

Let $(X,||\cdot||)$ be a normed space such that, for $x,y\in X$ $$||x + y||^2 +||x - y||^2 = 2||x||^2 + 2||y||^2$$. Then I want to check that $||\cdot||$ is induced by an inner product, so what I ...
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3answers
65 views

Why aren't the rationals a compact subset of $\mathbb{R}$?

We define a compact subset of some normed vector space $V$ to be any subset $S$ where every sequence $\{\mathbf{x}_{n}\}$ in $S$ has a subsequence which converges to some $\mathbf{x}$ in $S$. Then ...
0
votes
1answer
27 views

If $ T(f)=\int_0^12xf(x)\,dx$ find the value of $||T||$

Let , $T:(C[0,1],||.||_{\infty})\to \mathbb R $ be defined by $\displaystyle T(f)=\int_0^12xf(x)\,dx$ for all $f\in C[0,1]$. Then find $||T||$, where , $\displaystyle ||f||_{\infty}=\sup_{0\le ...
2
votes
2answers
43 views

Separability of $l_p(I,K)$

Good day, I have the following question "Prove that for $1 \leq p < \infty$, $l_p(I,K)$ is separable if and only if $I$ is countable, and $l_{\infty}(I,K)$ is separable if and only if $I$ is ...
0
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1answer
69 views

On continuous mappings on closed unit balls which is injective in the interior

Let $f: B[\theta,1] \to B[\theta , 1]$ be continuous and is injective in $B(\theta , 1)$ ; then is it true that the set $\{x \in Bd \space B(\theta,1): |f^{-1}(\{x\})|\ge3\}$ is countable ? (here ...
0
votes
2answers
32 views

How does one prove that two norms are equal if and only if their closed 1-balls are equal?

Let $X$ be a vector space and let $||\bullet||_1$ and $||\bullet||_2$ be two norms on $X$. I wish to prove that $||x||_1=||x||_2$ for all $x\in X$ if and only if $\{ x\in X \text{ such that ...
1
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0answers
40 views

Continuous linear bijection of a Banach space is a homeomorphism

I have seen an example of a continuous linear bijection $f:S\to S$, where $S$ was a normed linear space, such that the inverse function $f^{-1}$ was not continuous,as it was unbounded.The norm on $S$ ...
2
votes
3answers
159 views

$T$ is surjective if and only if the adjoint $T^*$ is an isomorphism (onto its image)

I am trying to prove the following statements: Let $X$ and $Y$ be normed spaces (not necessarily complete) Let $T\in L(X,Y)$ (meaning $T:X\to Y$ is a bounded linear map). Let $T^*:Y^*\to X^*$ denote ...
0
votes
1answer
42 views

Is there any continuous function $f:D^n \to S^{n-1}$ whose restriction to the sphere is the identity?

Is there any continuous function $f:D^n \to S^{n-1}$ whose restriction to the sphere is the identity ? If there does not exist such a function then can we deduce Brouwer fixed point thoerem from this ...
5
votes
0answers
68 views

On the form of injective function $f:\mathbb R^+ \to \mathbb R^+$ such that $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ is a norm

Let $f:\mathbb R^+ \to \mathbb R^+$ be injective such that on $\mathbb R^n , n>1)$ , $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ , where $u=(u_1,u_,\ldots,u_n)$ , defines a norm ; then is it true ...
3
votes
1answer
47 views

Is every closed set , the set of zeroes (resp.critical points) of some smooth real valued function? [duplicate]

Let $A$ be a closed subset of $\mathbb R^n$ : 1) Is it true that for some smooth function $f: \mathbb R^n \to \mathbb R$ , $A=f^{-1}(\{0\})$ 2)Is it true that for some smooth function $f: \mathbb ...
0
votes
0answers
29 views

What does a norm which is symmetric “around” an index subset look like?

I am looking at norms $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ that have the following symmetry property $$\forall \beta \in \mathbb{R}^{n} \text{: }, \lVert\beta \rVert=\lVert ...