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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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1answer
23 views

Norm and unit sphere

I want to prove the following statement: If the unit sphere in a normed space contains a segment, then there exists two vectors $x,y$ such that $\Vert x + y \Vert = \Vert x \Vert + \Vert y \Vert$ ...
11
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2answers
584 views

When does an inner product induce a norm?

When we consider a vector space $V$ over some field $F$, I know that when the $F=\mathbb{R}$ or $ =\mathbb{C}$, by setting $\|x\|=\left\langle{x,x}\right\rangle^\frac{1}{2}$ we get a norm. However, ...
3
votes
1answer
28 views

Understanding completion of metric (vector) spaces

I am wondering if I have understood the consept of completion of a metric/normed space correctly. As I have understood the completion theorem, it is: $$\textbf{Completion theorem for metric spaces}$$...
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2answers
26 views

Give example of metric space /normed linear space in which we have a proper subset which is clopen.

In metric space the example of discrete metric space is very trivial as in discrete msp every subset is clased as well as open,now iaaue becomes in case of normed linear space.whether such a normed ...
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1answer
83 views

$\sup \left\| A x + B y + C z \right\|$ subject to $\left\|x\right\| = \left\|y\right\| = \left\|z\right\| = 1$

I'm interested in finding $\sup \left\| A x + B y + C z \right\|$ subject to $\left\|x\right\| = \left\|y\right\| = \left\|z\right\| = 1$ where $A$, $B$, $C$ and $x$, $y$, $z$ are real matrices ...
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1answer
15 views

how to define an inner product that inverse the original distance and have an associated norm

supposing that I have an inner product $\langle a,b\rangle$ based on the normed space. I want to inverse the distance, that is if the distance between two vectors are small, under the new inner ...
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0answers
43 views

Norm of identity operator

Let $1<p,q<\infty$, so I need to find a norm of identity operator $J: l^n_p \to l^n_q$ When $p\le q$, it's quite easy for me to understand that $||J||=1$ (beacuse in this case $||x||_q \le ||x||...
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0answers
21 views

Does the completion of an operator with closed image also have a closed image?

Let $X,Y$ be real normed vector spaces, and suppose that $T:X \to Y$ is a bounded linear operator with closed image. Let $\tilde X,\tilde Y$ be the completions of $X,Y$, and let $\tilde T:\tilde X \...
1
vote
1answer
49 views

Relation Between Norm and Inner Product

I have seen a proposition that Let $(X, \|.\|)$ is a normed space. If this norm can be induced by an inner product (i.e. it satisfies parallelogram property) then is induced by at most one inner ...
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0answers
31 views

Natural tramsformation between a normed linear space and it's double dual.

Let $X$ be a normed linear space. Then I know that $X$ is imbedded in it's double dual $X^{**}$ via the natural tramsformation $J : X \longrightarrow X^{**}$ (say). I have proved that $J$ is one-to-...
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2answers
54 views

Prove that Banach algebra $B(X)$ has a unique (up to equivalence) complete algebra norm.

Let $X$ be a Banach Space, and denote by $∥ · ∥$ the standard norm on $B(X)$, the space of bounded linear functions $T:X\to X$. (a) Suppose that $||| · |||$ is another algebra norm on $B(X)$. ...
2
votes
2answers
47 views

Is a linear operator continuous if its kernel is closed?

Let $T$ be a linear operator between two infinite dimensional normed spaces $X$ and $Y$ whose kernel is a closed subset of its domain. Does it imply that $T$ is bounded or not necessary ? If yes ...
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0answers
21 views

It is banach space or just normed space ? [duplicate]

Let $X$ be a nonempty set and $B(X,C)$ the set of all complex functions defined on $X$ and $\sup|f(x)|<+\infty$. Define norm $||\cdot||$ on $B(X,C)$ by $||f||=\sup|f(x)|$. Is $(B(X,C),||\cdot||)$ ...
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0answers
63 views

Normed spaces where $x, y \neq 0$, $\Vert x + y \Vert = \Vert x \Vert + \Vert y \Vert $ and $\forall c > 0, \ x \neq cy$

For vectors $x, y \in \mathbb{R}^n\setminus\{0\}$, under the Euclidean norm we have that $$\left\Vert x + y \right \Vert_2 = \left \Vert x \right \Vert_2 + \left \Vert y \right \Vert_2 \iff \exists c \...
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1answer
30 views

Why is the product of two norms is always bigger or equal to the norm of the same corresponding element? [closed]

E.g. if $A$ is a matrix and $v$ is a vector which can be multiplied with the matrix, it always applies that (no matter how the norm is defined): $||A|| ||v||\geq ||Av||$ Why is it so?
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0answers
24 views

Convergence in mean in a normed space

Let $a_1,a_2,\dots$ be random variables taking values in $\mathbb{R}^n$. Suppose they are i.i.d with mean zero. Given an arbitrary norm on $\mathbb{R}^n$, I want to know how fast $\mathbb{E}[\|\frac{...
1
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0answers
17 views

Weird inequality with $l^1$-norm

I need to prove the following. Set $$|\cdot|_1: R^n \rightarrow R, \;\; |x|_1 = \sum_{i=1}^n |x(i)|$$ and $$X = \{x \in R^n|\, 0 \leq x(i) \leq 1\; \forall i\}$$ Then $$|z|_1|x-y|_1( |x|_1 +\ |y|...
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0answers
49 views

Upper bound for $(AB-BA)x$

Given matrices $A,B\in\mathbb{R}^{n\times n}$where matrix $A$ is a diagonal matrix and $B$ is an upper triangular matrix. I'm looking for an upper bound for the expression \begin{align*} (AB-BA), \end{...
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1answer
36 views

Hahn Banach extension question.

Consider $\mathbb{R}^{2}$ with norm $||•||_\infty $ and let $Y= \{ (y_1,y_2) \in \mathbb{R}^{2} : y_1+y_2=0\}. $ If $g:Y \to \mathbb{R}$ is defined by $g(y_1,y_2)=y_2$ for $(y_1, y_2) \in Y$ then $(...
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0answers
44 views

A lower bound for the condition number matrix

I have the following proposition: Theorem: For every invertible matrix $A\in\mathbb{R}^{n\times n}$ and every matrix norm $\|\cdot\|$, then the condition number $\mathcal{K}(A):=\|A\|\cdot\|A^{-1}\|$ ...
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2answers
34 views

Can we approximate any eigenvalue of a matrix via eigenvalues of some sequence diagonalizable matrices which approximates the matrix?

Let $A \in M_n(\mathbb C)$ and $\lambda$ is an eigenvalue. Does there exist a sequence of diagonalizable matrices $D_n$ and a sequence $\{\lambda_n\}$ complex numbers such that each $\lambda_n$ is an ...
3
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0answers
57 views

Show that the linear operator is bounded and find its norm

Let $T:L^{1}(-\infty,\infty)\rightarrow C(-\infty,\infty)$ be an operator defined by $$(Tf)(x)=\int_{-\infty}^{x}f(t)\sin(t)dt$$ Show that is is bounded and calculate its norm. The space $C(-\infty,\...
1
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1answer
19 views

“$(x_n, y_n)$ is weakly convergent to $(x,y)$” implies “$(x_n)$ is weakly converging to $x$”?

Let $H_1$, $H_2$ and $H_1 \times H_2$ be three Hilbert spaces, let the sequence $(x_n, y_n)$ be weakly convergent to $(x,y)$ in $H_1 \times H_2$. Then, when do we have " the sequence $(x_n)$ is weakly ...
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0answers
50 views

Operator norm of $T: L^{p}(0,\infty) \rightarrow L^{p}(0,\infty)$

I hope you can help me with this exercise. Let $p\in (1,\infty)$. Also, \begin{align*} T: L^{p}(0,\infty) &\rightarrow L^{p}(0,\infty)\\ f&\rightarrow (Tf)(x)=\frac{1}{x}\int_{0}^{x} f(y)\, ...
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0answers
19 views

Uniform continuous and bounded set of functions is closed under scalar multiplication

$f(x)\in C_{ucb}$ (Uniformly continuous and bounded functions) Show $cf(x)\in C_{ucb}$ Would it be enough to write: $\forall \epsilon>0$; $ \exists \delta>0$ such that $|x-y|<\delta \...
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0answers
33 views

Show that the linear functional is bounded and find its norm.

Let us define a functional on a space of convergent sequences $c$ with a norm $\left\Vert x \right\Vert_\infty =\sup_{1\leq k\leq\infty}\left|x_{k}\right|$ by an equation: $$f(x)=\lim_{n\rightarrow\...
1
vote
2answers
44 views

Show $T$ is a bounded linear operator

Let $T:C[0,1]\rightarrow C[0,1]$ be defined as: $$(Tf)(x)=\int_{0}^{1}xyf(y)\,\mathrm{d}y.$$ Show $T$ is a bounded linear operator and calculate its norm. My idea: First I know that $\Vert f\Vert=\...
1
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1answer
61 views

A norm in the $C^1$ space

How do I prove that a norm defined by $$\left\Vert f\right\Vert =\left|f(0)\right|+\int_{0}^{1}\left|f'(x)\right|dx$$ is a norm in the $C^1[0,1]$ for functions of class $C^1$?
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0answers
22 views

norm on the space of continuous paths

Let $B$ be a Banach space and $C([0,1],B)$ be a space of continuous paths connecting 0 to $b \in B$. It seems that with the norm $\displaystyle |l| = \max_{t \in [0,1]} l(t) $ it becomes a Banach ...
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1answer
27 views

Let $(X, \| \|)$ be an NLS, $x\in X$ and $0<r<s$. Show that, $B(x,r)\subsetneq B(x,s)$.

It is an exercise given in my text book of "Topology of Metric spaces by S. Kumaresan" Note that since in an Norm Linear Space $B(x,r)=x+rB(0,1)$, if we can prove that $B(0,r)\subsetneq B(0,s)$ for $...
2
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4answers
29 views

Locally compact metric space having atleast one non-compact closed ball

Can we have a non-trivial example of a locally compact metric space in which atleast one non-trivial closed ball is not compact. I am considering that infinite set with discrete metric is a trivial ...
1
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1answer
30 views

Convergence pointwise of limited operators, then the sequence of norms is limited

I'm studying funcional analysis and our professor left the following problem: Let $E$, $F$ be normed spaces, and $\left( A_n\right)$ be a sequence of limited linear operators from E to F, such that, ...
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2answers
26 views

How to get remaining unit normals by purely geometric reasoning instead of calculating?

Here "~" represent the vector. Draw the tetrahedron AOBC through vertices A(1; 0; 0), the origin O(0; 0; 0), B(0; 2; 0) and C(0; 0; 3) in the standard cartesian 3D-frame. Calculate the outward unit ...
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1answer
56 views

To show the two metrics induce the same topology but only one of them is complete

We have a standard metric $d_1(x,y) := |x-y|$ on $\mathbb{R}$, and another metric $d_2(x,y) := |f(x)-f(y)|$ on $\mathbb{R}$, where $f(x)=\frac{x}{1+|x|}$. I want to show these two metrics induce the ...
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0answers
18 views

Converge in norm and converge point wise are not equivalent.

I have known that for sequences of functions in $C[0,1]$, converge point wise is not equivalent with converge in norm for that two special norms: Integral and Suprem. However, when it comes to any ...
2
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1answer
33 views

Norm in $\mathcal{C}[\Bbb{R},\Bbb{R}]$ [duplicate]

I’m interested in expliciting a norm in the space of continuous functions from $\Bbb{R}$ to itself. It does not need to induce a complete metric.
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1answer
31 views

Prove that $A$ is Banach space under the norm

Let $A= \left\{ \begin{bmatrix} 0 & a_{12} & a_{13} \\ 0 & 0 & a_{23} \\ 0 & 0 & 0 \\ \end{bmatrix} \bigg|a_{12},a_{13},a_{23} \in \mathbb{C} \right\}$. Now define $\|A\|=\...
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1answer
41 views

Functional Analysis Banach space.

Let $X=C^{1}[0, 1]$. For each of $f\in X$, define $$p_1(f):= \sup\{{|f(t)|:t\in [0, 1]}\}$$ $$p_2(f):= \sup\{{|f'(t)|:t\in [0, 1]}\}$$ $$p_3(f):=p_1(f)+p_2(f)$$ I know that $(X, p_1 ), (X, p_2 )$ ...
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1answer
35 views

Show that $d(x+y,A+B) \leq d(x,A) + d(y,B).$

Problem: Let $(X,d)$ be a metric space. For each nonvoid subsets $A$ of $X$ let $$d(x,A) = \inf_{y \in A} d(x,y), \quad x \in X.$$ Show that: $d(x,A) = d(x,\bar{A})$. In particular, $d(...
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1answer
9 views

Closed set in a finite-dimensional normed space.

A finite-dimensional normed space X is complete. Every complete space is also closed. If I take linearly independent $x_1,\dots,x_n \in X$, then the set $\{\sum_{i=1}^{n} a_i x_i : a_i > 0\}=:A$ is ...
1
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0answers
22 views

Try to find a seminorm $p$ on a locally convex space st $|f|\leq p$.

Let $X$ be a locally convex space. Let $M$ be a linear subspace of a locally convex space $X$. Let $f\in M^*$. Then can we find a seminorm $p$ on $X$ such that $|f(x)|\leq p(x)$ for all $x\in M$? ...
2
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1answer
51 views

What conclusion can we get from a closed unit ball of a subspace $M$ of a norm space $X$ is closed in $X$?

Let $X$ be a normed space and $M$ be a linear subspace of $X$. Let $B_M$ be a closed unit ball in $M$ such that $B_M$ is closed in $X$. Then can we get $M$ is closed in $X$? Or other conclusions we ...
3
votes
3answers
50 views

Is the normed linear space $X$ isometrically isomorphic to $Y$, if there is a linear operator $T: X \to Y$ such that $\|T\|=1$?

My question is if $\|T\|=1$ a sufficient condition for isometric isomorphism. If yes, how to prove it? If not, which additional conditions are needed? Perhaps $\|T^{-1}\|=1$ is necessary?
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0answers
9 views

Proving a proposition about special functions in finite dimensional Hilbert space

We call frame function any mapping $f : \{ x \in H, \| x \|=1 \} =:\mathbb{S}(H) \mapsto \mathbb{R} \cup \{ \pm \infty \}$ (here $H$ denotes a finite dimensional Hilbert space) satisfying the ...
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2answers
56 views

If $X$ is a non empty closed subset of $\mathbb{R}$ and $Y = [1,2]$ then $X+Y$ is closed.

If $X$ is a non empty closed subset of $\mathbb{R}$ and $Y = [1,2]$ then $X+Y$ is closed? How to show this? I know that sum of two closed set may not be closed. But how to prove this? Can anyone ...
1
vote
0answers
71 views

A proof of Banach-Steinhaus theorem.

My instructor has given me a proof of Banach-Steinhaus theorem which is also known as Uniform Boundedness Theorem which I find hard to grasp. Here it is $:$ Uniform Boundedness Theorem $:$ ...
1
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1answer
46 views

Example of a metric on $\mathbb{R}^2$ that is not induced by any norm.

From looking at the properties of both what makes a function a metric and a function a norm, I'd gather that I'd have to create a metric that would not satisfy the scalar multiplication property of a ...
0
votes
1answer
35 views

Is it true that $4$ distinct points in the plane form a rhombus if and only if the set of their distances contains exactly $2$ or $3$ elements?

Is it true that $4$ distinct points in the plane form a rhombus if and only if the set of their distances contains exactly $2$ or $3$ elements? The problem can be stated formally as: Let $I=\{1,2,3,...
2
votes
0answers
34 views

Composition of two linear continuous functions

Let $E,F, G$ be three Banach spaces and let $(v_n)$ be a sequence of continuous linear functions from F to G which converge to $v ,$ and $(u_n)$ be a sequence of linear and continuous functions from $...
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0answers
21 views

Existence of 1-complemented hyperplanes in finite dimensional normed spaces

Let $X$ be a $d$-dimensional normed space and $M$ a subspace of $X$, then we say that $M$ is 1-complemented if there exists a linear projection $P :X \rightarrow X$ such that $P(X)=M$ and $\|P\|=1$. ...