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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9
votes
2answers
380 views

Non-Banach, completely metrizable normed vector space

Does there exist a normed vector space $(X,\|\cdot\|)$ over $\mathbb R$ or $\mathbb C$ such that the metric induced by the norm $(x,y)\mapsto\|x-y\|$ is not complete; but there exists some other ...
1
vote
1answer
30 views

What is the generalized form of this identity and how to interpret it?

I have learnt that for any inner product space of $\mathbb{C}$, we have $$\langle f,g\rangle=\frac{1}{4}\Big[||f+g||^2-||f-g||^2+i\big(||f+ig||^2-||f-ig||^2\big) \Big]$$ I know how to prove it, but ...
2
votes
2answers
141 views

On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
1
vote
1answer
59 views

When a metric space is a normed space?

I'm trying to figure out that which condition should be provided for a metric space to be normed also?
2
votes
0answers
26 views

Separable $X^*$ Property

Let $X$ be a normed space. If $X^*$ is separable, then there exists $(f_n)_{n\geq1}\subset X^*$ such that $\|f_n\|_{X^*}=1$ for all $n$ and $\{f_n:n\geq1\}$ is dense in $\{f\in X^*:\|f\|=1\}$. In ...
1
vote
0answers
37 views

Isometric isomorphisms between normed spaces and compact hausdorff spaces

Let $X$ be a normed space. Show that there is a compact Hausdorff space $Y$ such that $X$ is isometrically isomorphic to a subspace of $C(Y)$. I think this might be proved using the Banach–Alaoglu ...
0
votes
1answer
33 views

Compute the following norm derivatives

I was wondering if anyone can explain me how to compute the derivatives of the following norms: $\frac{d}{ds}||x+sp||^2_q$ for $x,p\in\mathbb{R^n}$ and $1<q<\infty$ $\bigtriangledown ...
2
votes
1answer
34 views

Are the normed spaces $ \mathbb{R}^{n^2} $ and $ M_n(\mathbb{R}) $ isometric?

Consider the spaces $ \mathbb{R}^{n^2} $ with euclidean norm and $ M_n(\mathbb{R}) $ of $n\times n$ matrices with the norm defined by $ \Vert A\Vert = \sup\limits_{\Vert x\Vert \le 1}\Vert Ax\Vert$. ...
1
vote
0answers
60 views

Can a Norm be Induced by two Different Complex Inner Products?

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$. If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and ...
1
vote
0answers
36 views

Normed space of dimension $> 1$

Let $X$ be a normed space of dimension > 1. Prove that each $\epsilon > 0$ exist $x,y \in X$ such that $\left \| x \right \| = \left \| y \right \| = 1 $ and $ 0< \left \| x-y \right \|< ...
0
votes
1answer
42 views

normed space of infinite dimension

By $X$ we denote an infinite-dimensional normed space. Show then that exist $x \in X$ and a closed subset $F\subset X$ such that $d(x,F) < \left \| x - y \right \|$ for all $y \in F$. (Note: It ...
5
votes
3answers
70 views

A direct proof on if $(X, ||\cdot ||)$ is a normed vector space and $Y\subset X$, with $Y$ having finite dimension, then $Y$ is closed.

I am trying to produce a direct proof on the statement mentioned above. The field I am working in is $\mathbb{R}$. My proof outline goes as following: If $Y$ is finite-dimensional, there exists a ...
0
votes
0answers
20 views

Normed spaces welfare problem

A producer own a technology for transforming an input into an output. If x $\in$ $\mathbb{R}_+$ units of the input are employed, the technology yields at most an amount √x ∈ R+ of output. Let F := ...
0
votes
2answers
46 views

Convex hull, compactness, normed spaces

Let $(X,\| \cdot \|)$ be a finite dimensional normed space. Show that if $S\subseteq X$ is compact, then the $\text{Conv(S)}$ is also compact. I used the Caratheodory's theorem to show that ...
1
vote
1answer
29 views

Is there a bounded dense subset of norm linear space?

I have a question. In norm linear space $X$, we can find a bounded dense subset of $X$, can´t we?
0
votes
1answer
55 views

Normed space and convex hull of closed subset

Let $(V, ||\cdot||)$ be a normed space. If $ C\subseteq V$ is a closed set we do not know if $ch(C)$ is closed or not. The professor provided this example that as of now I'm not getting: Consider the ...
0
votes
1answer
46 views

Some Strange Minimum and proof

I read following on Norm chapter in one book. $$\begin{align}\left|\left\|x-y\right\|-\left\|w-z\right\|\right| \leq & \min \{\|x-w\| + \|y-z\| , \|x-z\| + \|y-w\|\}\\\text{ or, }&\min ...
0
votes
1answer
66 views

Space of real polynomials of one variable isn't complete

Consider $E$ - the vector space of all real polynomials of one variable. I need to prove that it is not complete under any norm. I was thinking I could use the fact that certain functions, for ...
2
votes
1answer
150 views

Why isn't $\,\mathcal C[0,1]$ a Banach space in this unusual norm?

I wish to ask the following question: Let $\mathcal X$ be the normed space $\,\mathcal X=\mathcal C([0,1])$, with norm defined as $$ \|\,f\|= \max_{x\in[0,1]} x^2 \lvert\,f(x)\rvert. $$ Why isn't ...
0
votes
1answer
55 views

Norms for which every subset of closed unit ball containing the open unit ball is convex

It can be shown without much difficulty that any Euclidean norms satisfies the following condition :$$(P) \quad B \subset X \subset B' \Rightarrow X \, \text{is convex}$$ where $B=\{x \in E / \|x\| ...
0
votes
1answer
41 views

Continuity and norm of a functional

Let $E = \mathbb{R} [X]$ equipped with the norm $||p|| = \int_0^1 (|p(t)| + |p'(t)|) \ d t $. Check if the functional $\psi : E \ni p \rightarrow p(0) \in \mathbb{R}$ is continuous, and if it is, ...
-1
votes
1answer
44 views

Prove some Equivalences Norm

Suppose $X=R^2$ and $x=(x_1, x_2)$. I see the following are equal EDIT: ( equivalence). why? i couldent find any proof to satisfy me. any hint or idea or proof highly appreciated. $||x||_1= |x_1| + ...
0
votes
1answer
39 views

Unit Ball in 1 norm is open in ($C[0,1] , || \quad ||_{\infty}$)

Claim $B_1(0,1) := \{ f \in C[0,1] ; ||f||_{1} < 1 \} $ in $(C[0,1],||\quad || _1)$ is open in $(C[0,1],||\quad || _{\infty}).$ We need to take any $f \in B_1(0,1),$ and we have to find an ...
2
votes
0answers
141 views

Completely metrizable space, complete norm

Let $E$ be a normed, completely metrizable space. Prove that the initial norm is complete. How can I go about solving this problem? I will be grateful for all your hints. Thank you!
0
votes
2answers
255 views

Norm equivalence (Frobenius and infinity)

I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius. I have managed to solve find the constants for $||.||_{1}$ and $||.||_{2}$ but I cannot see how to continue ...
2
votes
1answer
50 views

How to compute the induced matrix norm $\| \cdot \|_{2,\infty}$

The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|_p)$ to $(\mathbb R^n, \| \cdot \|_q)$ is given by $$ \| A \|_{p,q} = \sup_{x\in\mathbb{R}^n\setminus \{0\}} ...
0
votes
1answer
111 views

Lipschitz continuity and gradient of a real-valued function on a normed space

The gradient of a function $f:E\to\mathbb{R}$ is Lipschitz continuous with parameter $L > 0$ iff $$\|\nabla f(x) - \nabla f(y)\|^* \le L\|x-y\| \quad \forall x,y\in E.$$ I have two questions: ...
0
votes
1answer
47 views

Show that a subspace of a normed vector space is closed

Let $X$ be a normed vector space over $\mathbb K, \mathbb K = \mathbb R$ or $\mathbb K=\mathbb C.$ Let $Y$ be a closed linear subspace of $X$ and $x\in X\backslash Y.$ Set $Z=\{y+\alpha x;\;y\in ...
1
vote
1answer
49 views

Prove that a subset of an infinite dimensional complete space is uncountable

Let $X$ be a Banach space. A subset $S ⊂ X$ is called a Hamel basis of $X$ if $S$ is linearly independent and every element of X is a finite linear combination of elements of $S$. (i) Prove that if ...
3
votes
1answer
70 views

Computing the norm of a linear operator

For two finite-dimensional real vector spaces $E_1,E_2$, define an linear operator $A:E_1\to E_2^*$. Its adjoint operator is defined by $A^*:E_2\to E_1^*$ its adjoint operator, i.e. $$\langle Ax,u ...
1
vote
2answers
35 views

$C^n[a,b]$ as a normed algebra

I would like to prove that the space of complex valued functions, differentiable $n$ times with continuous derivative, $C^n[a,b]$, with the metric defined by the norm ...
0
votes
1answer
52 views

When Heine - Borel theorem holds

If $\cal A$ is an abelian Banach algebra, then its maximal ideal space $\Omega$ is a compact Hausdorff space. In the proof of this theorem, the author says, since $\Omega \subset ball \cal A^*$, it ...
3
votes
1answer
116 views

An open set in the space of bounded real sequences

Let $X$ denote the set of all bounded real sequences, equipped with the norm $\| (x_n)\|_\infty:= \sup\{|x_1|,|x_2|,|x_3|,\ldots\}$; Let $X_{++}$ denote the set of all bounded positive real sequences ...
1
vote
1answer
84 views

Example of a proper, dense ideal in a unital Banach algebra

I know that if ${\cal A}$ is a non- unital Banach algebra, then ${\cal A}$ may contain some proper, dense ideals. I need an example of a proper, dense ideal in a unital Banach algebra or show that it ...
0
votes
1answer
45 views

Extend Metric Space Challenge

Let $(E, D)$ be a metric space. Consider $D_1: E\times E \to \mathbb{R}$ where $$ D_1(x,y)=\frac{D(x,y)}{1+ D(x,y)}. $$ I read some note about it but I want to find why $D_1$ is also a metric and ...
1
vote
1answer
71 views

Show that $\infty$-norm and $C^1$-norm are not equivalent.

Show that $\infty$-norm and $C^1$-norm are not equivalent. For the $C^1([a,b],\mathbb{R})$ space, show that $\displaystyle ||g||_\infty=sup_{a\leq t\leq b}|g(t)|$ and $\displaystyle ...
0
votes
2answers
26 views

Are lines in arbitrary normed vector spaces closed?

Let $(V, \| \cdot \|)$ be a normed (real) vector space. Given two vectors $a$ and $d$ (with $d$ not the zero vector), is the line $ L = \{a + td: t \in \mathbb{R}\} $ through $a$ in direction $d$ ...
5
votes
2answers
1k 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, ...
3
votes
1answer
46 views

Convergence of $a_n=d(u_n,\Bbb{Z})$ where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$.

I would like to study the sequence defined by $$ a_n=d(u_n,\Bbb{Z}). $$ Where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$ and $d(u_n,\Bbb{Z})=\inf\{d(u_n,x):x\in\Bbb{Z}\}$. I do not really ...
2
votes
1answer
72 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 ...
1
vote
1answer
37 views

Prove that for $0<p<1$, $|x-y|^p$ is a metric space on $\mathbb R^{n}$

Define the function $f_p : \mathbb R^{n} \to \mathbb R^{n}$ for $n ≥ 2$ by $f_p(x) = \sum_{k=1}^{n} \lvert x\rvert^{p}$. Show that for $0<p<1$, we get $d_b(x,y) = f_p(x-y)$ is a metric on ...
0
votes
1answer
35 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
93 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
93 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
52 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
38 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
97 views

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

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
40 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
111 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 $A$ ...
0
votes
0answers
68 views

Inner product spaces, normed spaces, metric spaces and topological spaces

I am collecting theorems or properties that hold in IPS, NS, MS or topological spaces, but not all of them. The reason is that I want to create some sort of overview over the respective spaces and ...