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Questions tagged [normed-spaces]

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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Prove the continuity of a function in a normed space

Let $(X,||·||)$ be a normed space over a field $\mathbb{K}$, where $$||x||=\max\{||x_1||,||x_2||\}$$ for any $x=(x_1,x_2)\in\mathbb{K}\times X$. Show that the mapping \begin{equation*} \begin{...
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+100

If the dual of a topological vector space separates points, does it separate a point and a closed subspace?

The Hahn-Banach Theorem implies that if $X$ is a normed vector space, then the dual space $X^*$, consisting of continuous linear functionals on $X$, has the following two properties: $X^*$ separates ...
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17 views

Does the dual space of a normed vector space separate closed sets?

If $X$ is a normed vector space, then the dual space $X^*$, consisting of continuous linear functionals on $X$, separates points. What that means is that if $x_1,x_2\in X$, then there exists an $f\in ...
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2answers
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Prove that $\Vert T \Vert =\sup\limits_{\Vert x \Vert< 1}\Vert T x \Vert,\;\forall\;T\in B(X,Y).$

Prove that $\Vert T \Vert =\sup\limits_{\Vert x \Vert< 1}\Vert T x \Vert,\;\forall\;T\in B(X,Y).$ Trial It suffices to show that $\sup\limits_{\Vert x \Vert< 1}\Vert T x \Vert=\sup\limits_{\...
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2answers
26 views

Why is $\|x+y\|^2=4 \| (x+y)/2 \|^2$ in Normed space?

Why is $\|x+y\|^2=4 \| (x+y)/2 \|^2$ in normed space? This could be an application of parallelogram law, but I don't see how.
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1answer
16 views

Must a Convergent Net in a Normed Space be Bounded?

If $ X $ is a normed space and $ (x_n)_{n=1}^{\infty} \subset X $ is a convergent sequence, then it is elementary to show that $ \| x_n \| $ is bounded by observing that there exists an $ N \in \...
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1answer
39 views

Normed space $C^2[0,1]$ with norm $\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$ is Banach space

The problem is as follows: I want to show that the normed space $C^2[0,1]$ with norm defined as $$\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$$ is a Banach space (and I ...
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Is $0$ the only vector in the kernel of every bounded linear functional?

Let $X$ is a normed vector space, and let $x_0\in X$ have the property that for every bounded linear functional $f:X\rightarrow K$, $f(x_0)=0$. Then does $x_0=0$? I think the answer is clearly yes, ...
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1answer
27 views

Showing weakly continuous operators are continuous without using weak topology

Let $X$ and $Y$ be Banach spaces, and let $T:X\rightarrow Y$ be a linear map such that $f\circ T$ is continuous for all $f\in Y'$. Show that $T$ is continuous. Now I think this problem is trivial ...
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1answer
33 views

Is the addition and scalar multiplication continuous with respect to this particular topology?

Suppose $X$ is a normed linear space. If $\mathcal {B}$ consists of $\phi$, $X$, all open balls centered at origin, and all open annulus centered at origin, then it is clear that $\mathcal {B}$ is a ...
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Derivative Normed Vector Spaces

Could someone provide an explanation for Proposition 3.9 below. The author states "The proposition is a simple application of Theorem 20.6" where Theorem 20.6 is the chain rule. However, I am ...
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37 views

Show that A,B are closed but A+B is not ( for given example of A and B)

X is a normed linear space and A,B be subsets of X. Prove that, (a) A+B is open if A is open or B is open. (b) A+B is closed if A is closed and B is compact. (c) Consider $X=c_0$, $A=\{(...
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Matrix norm if we don't specify the vector norm

Is it possible to compute the norm $$ ||A||=\sup_{X\in \mathbb{R}^n\setminus \{0\}} \frac{\lVert AX\rVert}{\lVert X\rVert} $$ where $A$ is the matrice defined by $AX=(x_1-x_n,x_2-x_1,\ldots,x_n-x_{n-1}...
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1answer
34 views

The spaces $\ell^p, \; 1 \leq p < + \infty$ are separable. On the other side, $\ell^\infty$ is not.

Exercise : Show that the spaces $\ell^p, \; 1 \leq p < + \infty$ are separable. Attempt : In order to show that $\ell^p$ is separable for $\ell^p, \; 1 \leq p < + \infty$, we need to work ...
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3answers
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Banach Spaces, Convergence and Spectrum

Let $V$ be a Banach space and $T_n → T$ in $B(V)$. Assume $λ_n ∈ σ(T_n)$ and $λ_n → λ$, I want to show that $λ ∈ σ(T)$. Okay, so if $\lVert T_n-T\rVert_{\mathcal B(V)}\to 0$ and $\lambda_n\to \...
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2answers
33 views

Unit ball of $X^{**}$ is weakly compact!

Is it true that the closed unit ball in $X^{**}$ is compact with respect to the weak topology on $X^{**}$, where $X$ is a Banach space? If so, how can we prove it?
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0answers
32 views

Grothendieck's double limit criteria for weakly compact set

I was reading about Arens regularity of normed algebras in Palmer's book. He talks about the Grothendieck's double limit criteria for relatively weak compact sets in $X^*$ . Can some one provide me a ...
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1answer
56 views

Continuity of a function (upper envelope)

Suppose $n,m\geq 1$ and let $A\subset \mathbb{R}^m$ compact. Let $f:\mathbb{R}^n\times A\to \mathbb{R}^n$ and $l:\mathbb{R}^n\times A\to \mathbb{R}$ such that $f$ is continuous and there exist $L_f,...
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1answer
33 views

Is $(\ell_1 , \Vert \cdot \Vert_2)$ a complete space?

I know that $\ell_2$ with respect to $\Vert \cdot \Vert_2$ norm is complete? I can't figure out for this set.
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1answer
17 views

Proposition regarding inverse operators and their norm

I'm trying to prove the following assertion. Hypothesis: $1)$ $X$ is a Banach space and $Y$ is a normed space. $2)$ $A : X → Y$ is a bounded bijective operator and $A^{-1}$ is bounded. ...
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Norms in normed spaces are always assumed to be finite? But where are infinite norms then?

Norms in normed spaces are always assumed to be finite? True? But where are infinite norms then? This is the "background" that I have been presented as to why questioning the finiteness of Cauchy-...
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2answers
27 views

How to use closedness of $C$?

Let $C$ be a closed subset of $\Bbb R^n$ and $r$ be a positive real. Consider the set $$ D = \{ y \in \Bbb R^n : \exists\ x \in C\ \text {such that}\ \|x-y\| = r \}.$$ Show that $D$ is a closed ...
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1answer
28 views

Is $D$ closed in $\Bbb R^n$? [closed]

Let $C$ be a non-empty closed subset of $\Bbb R^n$ and $C × D$ be a closed subset of $C × \Bbb R^n$. Can we say that $D$ is closed in $\Bbb R^n$?
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1answer
21 views

Confusion about real separable normed space problem

Suppose that $X$ is a real separable vector space, and $W$ a closed linear subspace of $X$. Show that there exists a sequence $(z_j )\in X$ such that $z_{j+1} \notin W_j := \mathrm{span} \ W \cup \...
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2answers
43 views

Norm in Sequence Space such that Convergence in Norm does not imply Pointwise Convergence

I'm wandering if there exist a norm defined on the sequence space $\mathbb{F}^{\omega}$ such that there exist a sequence that converges in norm but it doesn't converge pointwise, element by element. ...
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25 views

Radius, diameter and center of a set

I have 3 questions: Prove that for any set $A$ in a normed space $G$ we have: $r(A) \le d(A) \le 2 r(A)$, here $r$ is the radius of $A$, $d$ is the diameter of $A$. Find an example of a normed space ...
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1answer
24 views

Characterization of a rotund space

A normed linear space $X$ is said to be rotund if for all $x,y\in X$ with $\|x\|=1=\|y\|$, $\|x+y\|<2$. I want to prove that a normed linear space $X$ is rotund iff the function $\varphi:X\to \...
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4answers
515 views

Neither open nor closed subspace of a vector space?

Consider a vector space $V$ over $\mathbb{C}$ with some norm (and topology induced by that norm). I am trying to find a subspace $W \subset V$ such that it is neither open nor closed in the topology ...
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4answers
59 views

$X'$ finite-dimensional implies $X$ finite-dimensional

How would one prove, for any normed space $X$ that if $X'$ is finite dimensional, then $X$ is finite-dimensional? Here $X'$ denotes the space of all bounded functionals $f: X \to\mathbb F$ If anyone ...
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1answer
20 views

The extension of functionals

I'm facing the proof, using theorem of Hahn-Banach. The Theorem is following: In normed space every linear, continuous functional f on vector subspace $M \subset X$ can be extended to a continuous ...
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2answers
37 views

Is this linear operator on polynomials with sup-norm bounded?

Question: Let $\mathcal{P}$ be the space of all polynomials (with real coefficients) on the real line, endowed with sup-norm (i.e., $\|p\| = \sup_{0\le x\le 1}|p(x)|$). For any fixed $n \in \mathbb{N}...
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0answers
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An alternative way to show that any two norms on a finite dimensional vector space are equivalent. [closed]

I encountered this different method on page 432 of John Lee's Introduction to Smooth Manifolds. The hint in the book states that first choose an inner product on the vector space, and show that the ...
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1answer
31 views

Proving inequality involving scalar product on $\mathbb{R}^n$

Suppose $n,m\geq 1$, $A\subset \mathbb{R}^m$ is compact and $f:\mathbb{R}^n\times A\to \mathbb{R}^n$ is a continuous function satisfying $$\left\lVert f(x_1,a)-f(x_2,a)\right\rVert \leq L\left\lVert ...
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1answer
26 views

How this inequality is derived?

Let $T$ ∶ $ℓ_2$ → $ℓ_2$ be defined by $T((x_1,x_2,...,x_n...))$=$(X_2-X_1, X_3-X_2,...,X_{n+1}-x_n,...)$ Then I have find the norm of $T$. Here is the answer to this question: https://math....
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0answers
48 views

Is $P$ dense in $X$?

Let $C[0,1]$ denote all the real-valued continuous function on $[0,1].$ Consider the normed linear space $$X = \left \{f \in C[0,1] : f \left ( \frac 1 2 \right ) = 0 \right \},$$ with the ...
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1answer
17 views

Non-reflexive space that is isomorphic to its second dual space

I was wondering if it is possible to construct a space that is non-reflexive (so it is not isomorphic to its second dual space under the cannonical embedding), but some isomorphism exists between them....
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1answer
20 views

Showing the set of polynomials of degree $>n$ is open in the set of polynomials with norm $\|P\|=\max\{|p_0|,\dots,|p_d|\}$

Let $(X,\|.\|)$ be the normed vector space of polynomials with $\|p_0+p_1t+\dots+p_nt^n\|=\max\{|p_0|,\dots,|p_n|\}$ with normal addition and multiplication by a scalar. I have proved that $U_n=\{P(t)...
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1answer
27 views

Inequality of norms

Is it obvious that in $\mathbb R^n, \forall \ p,q \ \text{ s.t. } \ p \leq q$ $$ \vert \vert x\vert \vert_{p} \leq \vert \vert x\vert \vert_{q}, \text{where }\vert \vert x\vert \vert_{q} = (\sum_0^n ...
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1answer
63 views

Representation of linear operator between $L^p$ spaces.

I was wondering where I could find a reference to the a characterization of continuous linear operators: $$T:L^p(X,\mu)\to L^q(Y,\eta)$$ of the form $T(f)(y)=\int_{X} k(x,y)f(x)d\mu$ for some $k$ ...
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1answer
9 views

Non-constant function with 0 Lipschitz semi-norm

Suppose we have a bounded metric space $(X,d)$. We say a function $f:X\to \mathbb{R}$ is Lipschitz if $|f|=\sup_{\substack{x\neq y\\x,y\in X}}\frac{\left|f(x)-f(y)\right|}{d(x,y)}<\infty$. This ...
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1answer
34 views

A sequence of bounded $C^1$ functions whose derivatives are unbounded.

What is an example of a sequence of functions, $(f_n)_{n=1}^\infty\subset C^1([a,b])$, which are bounded in $C^1([a,b])$ under $\|\cdot\|_{\infty}$ but are such that their first derivatives $\|f'_n\|_{...
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1answer
30 views

Exist a norm such that $S= \{x= (x_{1},x_{2}) \in R^{2}| \|x\| \leq 1, x_{2} >0\}$ which isn't open nor closed?

Let $X = R^{2}$ and $\|.\|$ a norm in X: Show that exist a norm such that $S= \{x= (x_{1},x_{2}) \in R^{2}| \|x\| \leq 1, x_{2} >0\}$ it is not open or closed. How can I show this? Any help?
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2answers
63 views

$\Bbb R^d$ with norm is a Banach

Another problem I found in a textbook when brushing up on my real analysis. Was given a hint too. Prove that $\Bbb R^d$ with the norm $||x||_1=\sum_{i=1}^d |x_i|, \ x ∈ \Bbb R^d$ is a ...
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1answer
38 views

Closure of ball in normed space

I found this question in a text book and wanted to ask for some help. I am not quite sure where to start with this. Prove that in a normed space $\overline{B^o(x, r)} = B(x, r) \ \ \forall x ...
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1answer
35 views

Is the norm $p(x) = \max_{t \in [0,2]}|x(t)| + \left(\int_0^1|x(t)|^7\right)^{1/7}$ on $C[0,2]$ induced by any scalar product?

I have a norm on space $X = C[0,2]$: $$p(x) = \max_{t \in [0,2]}|x(t)| + \left(\int_0^1|x(t)|^7\right)^{1/7}$$ Is that norm induced by any scalar product? I try to find counterexample for ...
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1answer
23 views

understanding part of proof in Banach-Steinhaus theorem

Theorem: If a sequence of linear bounded operators $\{A_n\}_{n=1}^{\infty}$ is a Cauchy sequence in every point of the Banach space $E_x$, then the sequence of norms $\{\lVert A_n \rVert\}_{n=1}^{\...
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1answer
41 views

$T$ is closed $\iff$ for arbitrary $\{ x_n \}\in D(T)$ such that $x_n\to x,$ and $Tx_n\to y,$ we have $x\in D(T)$ and $Tx=y.$

Let $X$ and $Y$ be normed linear spaces and $T:X\to Y$ be any map. Then, $T$ is closed if and only if for arbitrary $\{ x_n \}\in D(T)$, domain of $T$, with $x_n\to x,$ and $Tx_n\to y,$ we have $x\in ...
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1answer
37 views

Continuity of Energy Functional

Let $u : \Omega \times [0,T]$ be a function such that $u \in C^{2,1}(\Omega \times [0,T])\cap C^{1}((0,T);L^{2}(\Omega))\cap C([0,T);H_{0}^{1}(\Omega))$ for $\Omega \subset \mathbb{R}$ an unbounded ...
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2answers
62 views

Proving the Lp norm is a norm.

I want to prove the $L_p$ norm on continuous functions is in fact a norm. I have proven definitiveness and homogeneity but am struggling with the triangle inequality. I am using the fact that if the ...
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1answer
119 views

Is $d(x,y)=\frac {\|x-y\|} {\sqrt {1+\|x\|^{2}}\sqrt {1+\|y\|^{2}}}$ a metric on a normed linear space?

Let $X$ be a normed linear space and $$d(x,y)=\frac {\|x-y\|} {\sqrt {1+\|x\|^{2}}\sqrt {1+\|y\|^{2}}}$$ Is this a metric? This question arose from the following post where the answer was given in ...