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
41 views

Using Nash inequality to derive an inequality (from proof in paper)

We work on a domain $\Omega \subseteq \mathbb{R}^N$ with the Dirichlet Laplacian. Let $\lVert \cdot \rVert_p$ denote the $L^p$ norm. I am trying to understand why the following inequality is true: ...
2
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2answers
59 views

Is a Normed Vector Space Necessary to Prove Path Connectedness?

Path connectedness seems to be defined in a topological space, but can the existence of a path be proven without using the functions of vector addition, scalar multiplication and norm ? For example, ...
0
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1answer
31 views

How to interpret these two equalities involving 2-norm?

What is the difference between $$\Vert x+y \Vert_2^2 $$ and $$\Vert x+y \Vert_2 $$ Can we write $$\Vert x+y \Vert_2 \stackrel{?}{=} \sqrt{\Vert x+y \Vert_2^2} \tag{*}$$ Moreover, when does (1) ...
2
votes
1answer
35 views

A normed space is Banach iff its unit sphere is complete [duplicate]

Let $X$ be a non-trivial (other than singleton $x$) normed space. Prove that $X$ is a Banach space if and only if $\{x \in X \mid \|x\| = 1 \}$ is complete.
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0answers
17 views

Does nonexpansive property in H-norm imply nonexpansive in 2-norm?

Suppose $\|f(x) - f(y)\|_H \le \|x - y\|_H$. In other words, $f$ is nonexpansive in the norm with respect to positive definite H: $\|z\|_H = z^T H z$. Can we then say something along these lines: ...
0
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1answer
14 views

Norm of pointwise product of Lp functions

Does the following inequality hold in $L_p$ spaces? $\|fg\|_p\leq\|f\|_p\|g\|_p$ How would I go about proving this? Do I need to apply Cauchy Schwarz?
2
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2answers
14 views

If $V$ is completely normable, then is every norm complete?

Here is a theorem that motivated my question. Let $(V,||\cdot||_V)$ be a normed space over $\mathbb{K}$. Then, there exists a Banach space $(X,||\cdot||_X)$ such that $V$ is dense in $X$ and ...
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0answers
25 views

Definition of James' Space

The definiton of James' Space in wikipedia begins with: Let $\mathcal{P}$ denote the family of all finite increasing sequences of integers of odd length. Shouldn't it be all finite increasing ...
1
vote
1answer
16 views

Non-separability of normed spaces

I would like some hints to decide when a normed space is separable or not. I really understood the definition and the classic examples of separable spaces but when I go to show that a space is ...
1
vote
1answer
18 views

Polarization Identity for Complex Scalars

So I was trying to prove that for $x,y\in \mathbb{C}$ we have that: $4 \langle x,y \rangle=||x+y||^2-||x-y||^2+i||x+iy||^2-i||x-iy||^2$. I got that $||x+y||^2-||x-y||^2=4\Re\langle x,y \rangle$ and ...
1
vote
1answer
52 views

How to proof that a finite-dimensional linear subspace is a closed set

Given a linear space V, a field F, a norm $||.||$ on V and a Base B. How do i proof that the sub-space span{$b_1,b_2,...,b_n$} where $b_i \in B$ is a closed set under the topology that is created ...
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0answers
19 views

Show that $Z$ is the null space of a suitable linear functional $f$ on $X$

If $Z$ is an $n-1 $ dimensional subspace of an $n$ dimensional vector space $V$ . Show that $Z$ is the null space of a suitable linear functional $f$ on $X$ which is uniquely determined upto a scalar ...
3
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1answer
16 views

Get the norm of a linear operator

Consider $C=\Big\{(z_n)\subset \mathbb{C}:\exists z\in\mathbb{C};\ z_n\to z\Big\}$ and $C_0=\Big\{(z_n)\subset \mathbb{C}: z_n\to 0\Big\}$ with the norm $||\cdot||_\infty$. If ...
0
votes
1answer
21 views

Show that $\exists $ a linear functional $f$ on $X$ such that $f(x_0)=1$ and $f(x)=0\forall x\in Z$.

Let $Z$ be a proper subspace of an $n$ dimensional vector space $X$ and let $x_0\in X-Z$. Show that there exists a linear functional $f$ on $X$ such that $f(x_0)=1$ and $f(x)=0\forall x\in Z$. ...
3
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0answers
65 views

Corollary of the Hahn-Banach theorem.

Recall the Hahn-Banach Theorem in Normed Vector Spaces: "Let $X$ be a NVS, and let $W$ be a linear subspace of $X$. Then, for any $f_W\in W'$ there exists an extension $f_X\in X'$ such that ...
2
votes
2answers
66 views

Prove that $I\colon(C[0,1],\|\cdot\|_\infty)\to(C[0,1],\|\cdot\|_1)$ is not an open map

Deduce that $I\colon(C[0,1],\|\cdot\|_\infty)\to(C[0,1],\|\cdot\|_1).$ is not an open map. This question has been resolved previously, but not by this way. I think that it's sufficient prove is ...
0
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0answers
25 views

$C^r(K,F)$ as a Banach space for $K$ compact, $F$ Banach space

Let $E$ and $F$ be Banach spaces and $K\subset E$ be compact. I want to understand what the "common definition" (if there is one) of the banach space $C^r(K,F)$ of $r$ times continuously ...
3
votes
1answer
45 views

Does $\|f\|_K = \sup_{z \in K} |f(z)|$ define a norm on $C(\mathbb{C})$ and $H(\mathbb{C})$ for compact $K$?

Let $C(\mathbb{C})$ denote the vector space of continuous complex valued functions on $\mathbb{C}$ and $H(\mathbb{C})$ denote vector space of entire functions. For any function $f$ in ...
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1answer
35 views

On the reflexivity of the $L^p$-spaces

If $X$ is a normed vector-space, then $X^\ast$ is the normed vector-space of bounded linear functionals $X \rightarrow \mathbf{R}$. Assume $1 < p, q < + \infty$ such that $\frac{1}{p} + ...
0
votes
1answer
33 views

How can I prove the following question

Let $(A,+,.,*,\|.\|)$ denotes complex Banach algebra such that $\|.\|$ norm on $A$ satisfies $$\|f*g\|| \leq \| f\|.\|g\|$$ and $e$ is the identity element. How can I prove that if $\| x\|<1$ ...
0
votes
1answer
16 views

Operator norm and continuity

I've read in the solution of an exercise: "$T$ has a finite norm, thus $T$ is continuous". We are in a normed vector space $(V,||.||)$ and $T$ is a linear selfmap over the vector space $V$. The ...
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0answers
20 views

If X* is separable X is also separable.X is normed vector space

I know the proof of this fact by contradiction. Is there any proof without contradiction or the reason why this happen? The Banach space $L(X,\mathbb{R})$ is called the norm dual of $X$ and is denoted ...
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1answer
32 views

Existence of sequence in $\ell^2$ - Uniform Bundedness Principle

Let $X$ be a vector space and $f_0,f_1,...:X\rightarrow \mathbb{K}$ linear functionals on $X$ such that for every $x\in X \ |f_0(x)|^2\le C\sum\limits_{n\in\mathbb{N}}|f_n(x)|^2<\infty$. Show that ...
4
votes
2answers
65 views

Show that this set is open in $E = C([0,1], \mathbb R)$, with the norm $||.||_\infty$

$E = C([0,1], \mathbb R)$, with the norm $||.||_\infty$. Let $O$ be an open of $\mathbb R$ and $$\Omega(O) = \{ f \in E: f(t) \in O, \forall t \in [0,1] \}$$ Show that $\Omega(O)$ is open in $E$ I ...
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vote
2answers
25 views

Denseness of vector space $V=C^1[0,1]$

Let $V=C^1[0,1]$, $X=( C[0,1],|| ||_\infty )$ and $Y=( C[0,1],|| ||_2 )$. Then $V$ is dense in $X$ but not in $Y$ dense in $Y$ but not in $X$ dense in both $X$ and $Y$ neither dense in $X$ nor ...
0
votes
0answers
15 views

Prove a problem on weak convergence [duplicate]

Given a normed space $X$, and $x_n, x\in X$, $x_n$ weakly converges to $x$. Prove: $x\in \overline{span\{x_n:n\geq 1\}}$. ($\overline{M}$ denotes the closure of $M$). I tried proof by contradiction: ...
3
votes
1answer
52 views

Prove that $\{x_n\}$ is bounded if $f(x_n)$ is bounded $\forall f\in X'$

Given a normed space $X$ and $x_n \in X$, for any $ f\in X'$ (dual space of $X$), $f(x_n)$ is bounded. Try to prove that $\{x_n\}$ is bounded. My thought is: if $\{x_n\}$ is not bounded, then for any ...
3
votes
3answers
56 views

How to show that the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not Hilbert space?

I want to show that the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not a Hilbert space. So I should show that it is not an inner product space. Most likely, The ...
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votes
0answers
25 views

Doubts about proof of completeness of finite-dimensional normed spaces

Every $n$-dimensional normed space $X$ (over $\mathbb R$ or $\mathbb C$) is complete. Let $\{e_1,\dots,e_n\}$ be a basis for $X$ and define the norm $\|\cdot\|$ on $X$ by $\|x\|=\max_{i\leqslant ...
1
vote
1answer
69 views

Two (equivalent ?) norms on Hilbert space

Let $H$ be a vector space equipped with two inner products $\langle \cdot,\cdot\rangle_1, \ \langle \cdot,\cdot\rangle_2$, s.th. $(H,\langle \cdot,\cdot\rangle_j)$ is a Hilbert space for $j=1,2$. ...
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votes
0answers
22 views

Convex and dense subset in infinite-dimensional normed vector space

Let $X$ be an infinite-dimensional normed vector space. How to construct proper convex subset $A$ of $X$, s. th. $A$ is dense in $X$ ? If $X$ is an unitary vector space then it is obvious, but how to ...
0
votes
1answer
16 views

Showing the the unit sphere is closed using sequences

Let $(X,\|\cdot\|)$ be a normed space. Prove that every sequence in $S_X=\{x\in X\mid \|x\|=1\}$ converges in $S_X$. My attempt. Let $(x_n)\in S_X$. Then, $\|x_n\|=1$ for all $n$. Now assume ...
0
votes
0answers
31 views

Is an unbounded function bounded on a bounded non-compact interval?

I'm a little confused about functions in the set of bounded continuous functions. For example, if we take the interval (0,1] and the function $f(x) = $\begin{cases} 0 & x \in ...
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vote
0answers
21 views

Proving Bellman operator being a contraction

I know that Bellman operator, defined as $T(f(x))=\sup_{y\in\Gamma(x)} {\phi(x,y)+\beta f(y)}$, is a contraction provided that $\beta\in(0,1)$ and $\phi$ is a bounded function on $Gr\Gamma$. In ...
0
votes
0answers
13 views

Nonisometric Minkowski spaces

Definition: A function $F:\mathbb{R}^{n}\to [o,\infty]$ is called Minkowski norm if it satisfies the following conditions: $F$ is $C^{\infty}$ on the ponctured space $\mathbb{R}^{n}-{0}$. $F(\lambda ...
0
votes
1answer
53 views

Proving that a given finite-dimensional vector space is isometrically isomorphic to $(\mathbb R^n,\|\cdot\|_\infty)$

Let $X$ be an $n$-dimensional vector space with a basis $\{e_1,\dots,e_n\}$. Consider the norm $\|\sum_{i=1}^n \alpha_ie_i\|=\max_{i\leqslant n} |\alpha_i|$ for $x=\sum_{i=1}^n\alpha_ie_i\in X$. We ...
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votes
1answer
46 views

Incompleteness of $\ell^1$ with respect to $\sup$ norm

I'm trying to make an example that shows $\ell^1$, that is the space of complex sequences that the sum of the norms of their components is finite, is not complete with respect to $\sup$ norm. And ...
4
votes
2answers
175 views

Space of Lipschitz Functions Complete?

Consider the subspace of continuous, real-valued functions on $[0,1]$ that are Lipschitz. Is this subspace complete under the sup norm ($\Vert \cdot \Vert_{\infty} = \sup \{ |f(x)| : x\in S \}$)? I ...
0
votes
1answer
24 views

If $T:X \rightarrow Y$ is a linear operator and $r>0$ such that $r \cdot B_Y \subseteq T(B_X)$, show $y ||x|| \leq M ||y||$.

Let $X$ and $Y$ be normed spaces and let $B_X$ and $B_Y$ denote the closed unit balls in $X$ and $Y$ respectively. Suppose $T:X \rightarrow Y$ is a linear operator and that there is an $r>0$ such ...
3
votes
1answer
52 views

Counterexample for the stronger statement of Riesz's lemma

Here is a counterexample for the stronger statement of Riesz's lemma and I don't understand it. Why for all $x$, such that $||x||=1$, there exists $y \in Y$, such that $d(x,y)<1$?
2
votes
2answers
50 views

Are these conditions also sufficient for a metric to be induced by a norm?

Let $(X,d)$ be a metric space such that the set $X$ is also a vector space over the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers. Then the following holds: If ...
1
vote
1answer
8 views

Find an example of an infinite dimensional vector space for which the following conditions of matrix convergence are not equivalent.

I need to find an example of linear operators $A_{n}$ and $A$ on an infinite dimensional vector space with norm $\| \cdot\|$ such that the following conditions are not equivalent: (i) $\|A_n -A\| \to ...
4
votes
1answer
92 views

Is there any popular name for this theorem in the standard literature?

Let $X$ be a normed space. Then $X$ is a Banach space if and only if the absolute convergence of any series in $X$ implies the conditional convergence of that series. Is there any name given to ...
0
votes
1answer
30 views

Is there a default norm for the (finite) product of Normed Vector Spaces?

I am trying to figure out what could be the norm associated with the product of two normed vector spaces. I know we can define several norms on it, for example, if we have $(X, \|·\|)$ and $(Y, ...
2
votes
3answers
47 views

Closure of a subset of normed vector space

Can you help me to prove this claim : $A$ is a subset of a normed vector space, closure of $A$ is closure of $$B=\bigcap_{n=1}^\infty \left( A+{1\over n}B_1 (0)\right)$$ I tried to prove ...
3
votes
0answers
40 views

$\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$

I'm learning about functions of bounded variations and need to verify my work this problem: Show that $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$. My attempt ...
2
votes
1answer
17 views

Let $E$ be a normed space and $b\in E$. Then $d(b, \overline{X}) = d(b,X)$.

$d$ is the distance between a point and a set: $d(b,X) = \underset{x\in X}{\inf}\{\|b-x\| \} $ and $X = B(a;r) = \{\|a-x\|<r: x \in E\}$, $ \overline{X} = B[a;r] = \{\|a-x\| \leq r: x \in E\}$ ...
2
votes
1answer
46 views

boundedness of convex functions

Let $X$ be a vector space, $\Omega$ a convex subset thereof and $f:\Omega \to \mathbb R$ a convex function. Then $f$ need not be bounded from below - not even if it is strictly convex, as the example ...
1
vote
2answers
29 views

Can derivative of a smooth norm be zero?

Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map. Is it true that its differential at (every non-zero point) ...
3
votes
1answer
46 views

When is the completion of a space of functions a space of functions?

If $V$ is a $\mathbb C$-vector space of functions $f: X \to \mathbb C$ on some common domain $X$ and $\tau$ is a Hausdorff, locally convex topology on $V$, when may the completion of $(V,\tau)$ also ...