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.

learn more… | top users | synonyms

0
votes
2answers
29 views

Show that there exist $\{x_n\}$ a sequence of elements in $A- \{x\}$ such that $x_n \to x$

Let $E$ a normed vector space and $A \subset E$. Let $x$ an accumulation point in $A$. Show that there exist $\{x_n\}$ a sequence of elements in $A- \{x\}$ such that $x_n \to x$. Definition : An ...
1
vote
1answer
26 views

Example of absolute convergent series divergent [duplicate]

I have learnt that in complete norm space any series that is absolute convergent is convergent. However, I am wondering is there any example of divergent series which is absolute convergent in that ...
3
votes
2answers
142 views

Can all vector spaces be made into normed spaces?

Can all vector spaces be made into normed spaces (even trivial ones)? Vectorspace could be of infinite dimension. Update: I don't know how to make this question more specific. I am talking about a ...
4
votes
2answers
248 views

Hahn-Banach Theorem for separable spaces without Zorn's Lemma

I was reading about the Hahn-Banach Theorem, its many versions and their proofs. It's known that in the proofs we need Zorn's Lemma. But in the book that I'm reading, the author said if $X$ is a ...
10
votes
2answers
188 views

Exists a uniformly convex norm on Banach space satisfying certain condition?

Let $E$ be a Banach space with norm $\|\cdot\|$. Assume that there exists on $E$ an equivalent norm, denoted by $|\cdot|$, that is uniformly convex. Given any $k > 1$, does there exist a uniformly ...
1
vote
0answers
26 views

$X$ is reflexive iff $X_R$ is reflexive

Problem: Let $X$ be a complex Banach space, $X_R$ its real version. Show that $X$ is reflexive if and only if $X_R$ is reflexive. My run at the solution: I suppose I should use the theorem, that ...
1
vote
1answer
49 views

To find the norm of a linear functional

Let $y\in C[a,b]$ and $f(x)=\int\limits_a^bx(t)y(t)dt$ for all $x\in C[a,b]$. I want to show that $f$ is bounded and $\|f\|=\int\limits_a^b|y(t)|dt$. I tried the problem as follows: ...
4
votes
1answer
79 views

To show $T$ bounded

Let $X,Y$ be normed linear spaces and let $T:X\to Y$ be a linear map such that for every absolutely convergent series $\sum\limits_{n=1}^{\infty}x_n$, the series $\sum\limits_{n=1}^{\infty}Tx_n$ ...
0
votes
0answers
33 views

Prove that $f(x,y)$ defines an inner product [duplicate]

Let $(E,\left\lVert . \right\rVert)$ be a normed vector space defined on $\mathbb{R}$ . We suppose that the norm satisfies the Parallelogram law. Prove that: $$f(x,y)=(1/4)[(\left\lVert x+y ...
2
votes
1answer
48 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
votes
2answers
79 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
votes
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
64 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.
0
votes
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
votes
1answer
28 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
votes
2answers
23 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 ...
0
votes
0answers
26 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
2answers
26 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
24 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
63 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 ...
0
votes
0answers
22 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
votes
1answer
18 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
46 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
votes
0answers
87 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
67 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
votes
0answers
26 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
46 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 ...
0
votes
1answer
41 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
35 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
20 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 ...
0
votes
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 ...
1
vote
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
68 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 ...
1
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
18 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
55 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
61 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 ...
0
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
82 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 $j=1,2$. ...
0
votes
0answers
24 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
18 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
35 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 ...
1
vote
0answers
38 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
14 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
54 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 ...
0
votes
1answer
47 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
228 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
63 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 ...