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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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Is there a special relationship between a norm on a vector space V, and the operator norm $ \mathcal{L}(V, \mathbb{R)}$?

Let $T$ be a linear operator in $\mathcal{L}(V)$. An operator norm is denoted as $||T||$, where it is the smallest $M$, such that $||T(v)||$ $\le$ $M||v||$ for any $v \in V$. A norm on the vector ...
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1answer
18 views

Norm of a bounded linear functional.

Let $X=(\mathbb R^2, \|.\|_3)$ be a real normed space, where $\|(x_1,x_2)\|_3=[|x_1|^3+|x_2|^3]^{1/3}$. How to find the norm of bounded linear functional $ax+by$? I tried this way: $|ax+by|\leq |a||x|...
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2answers
28 views

Normed linear space

In Walter Rudin's Complex Analysis, it states that by definition$$\|\Lambda\|=\text{sup}\{\|\Lambda x\|: x\in X, \|x\|\leq1\}$$ and then later he shows that $\|\Lambda x\|\leq \|\Lambda\|\|x\|.$ ...
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1answer
20 views

Prove that F is a bounded linear functional and $||F|| _{X^*}=||w||_{\infty}$.

Let $(X,||\cdot||)=(l^1,||\cdot||_1)$, $w=(w_n)_{n \geq 1} \in l^{\infty}$. Define for all $x=(a_n)_{n \geq 1}\in l^{1}$, $$F(x)=\sum_{n \geq 1}w_na_n.$$ Prove that $F$ is a bounded linear functional ...
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19 views

Uniform convexity requires stronger triangle inequality to be true uniformly.

From the wikipedia (in the section "Properties") https://en.wikipedia.org/wiki/Uniformly_convex_space : The strict convexity means a stronger triangle inequality $\|x+y\|<\|x\|+\|y\|$ holds ...
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1answer
39 views

A counterexample of Banach Steinhaus Theorem

I was reading about a consequence of Banach-Steinhaus theorem which states that: Let $E$ be a Banach space and $F$ be a normed space, and let $\{T_n\}_{n\in \mathbb{N}}$ be a sequence of bounded ...
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38 views

Show that $\sum_{k=0}^{N} |P(k)| \leq C(N) \int_{0} ^{1} |P(t)| dt $.

I’m attending a functional analysis course and I am given to solve this problem as an exercise but I’m a little bit disoriented and I don’t know what tools I can use to get it. Show that, for each $...
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1answer
22 views

$\limsup $ additivity. [on hold]

Let $x=[x_k , k\in\mathbb{N}]$ , And $y=[x_k , k\in\mathbb{N}]$ Two bounded sequences of $\mathbb{R^n}$. Do we have : $\limsup_k||x_k||+\limsup_k||y_k||=\limsup_k(||x_k||+||y_k||)$ ?
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Closed subspace and quotient norm

In the Banach space $C[0,1]$ consider the subspace $M=\lbrace g \in C[0,1]: \int_{0}^{1}g(t)dt=0 \rbrace $ Show that $M$ is closed in $C[0,1]$ and calculate the quotient norm $(\|f+M \|)$ where $f(t)...
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Operator in finite dimension spaces: $T: X \rightarrow Y$ is open iff $T(X)=Y$. [on hold]

To the functional analysis course I am attending I am given to solve this exercise. Given $X$ and $Y$ normed spaces, $Y$ with finite dimension. Show that a linear operator $T: X \rightarrow Y$ is an ...
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1answer
17 views

consequences of the uniform boundedness theorem

Let $X$ be the function space defined in the following way. A function $x$ belongs to $X$ iff the two conditions are satisfied: i) $x : \mathbb{R} \rightarrow \mathbb{R}$ ii) there exists a compact ...
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1answer
33 views

Closed and finite subspace [duplicate]

Given $X$ a normed space, $M$ a closed subspace of $X$ and $Z$ a finite dimension subspace of $X.$ Show that $M+Z$ is a closed subspace of $X.$ How can I do this? I thought to construct a sequence ...
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1answer
28 views

Are circles also squares in $(\mathbb{R}^2,||\cdot||_{\infty})$?

In $(\mathbb{R}^2,||\cdot||_{\infty})$ circles appear to be squares. But do squares exist in general normed space when we do not have an inner product, hence no natural notion of orthogonality and ...
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37 views

Functional Analysis and Norm Space property

Suppose $m(E)<\infty$ and $f\in\mathcal{L}^{\infty}(E)$ and $1\leq p_1<p_2\leq \infty$. The goal of the two problems below is to show \begin{align*} \lim_{p_1\to p_2}\|f\|_{p_1}=\|f\|_{p_2}....
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Prove that $(X,\Vert {\cdot}\Vert_2)$ is not complete where $\Vert f\Vert_2=\left(\int_{-2}^{2}|f(t)|^2 dt\right)^{1/2}$ [duplicate]

Prove that $(X,\Vert {\cdot}\Vert_2)$ is not complete where $X=C[-2,2]$ and \begin{align}\Vert f\Vert_2=\left(\int_{-2}^{2}|f(t)|^2 dt\right)^{1/2}.\end{align} MY TRIAL It suffices to produce a ...
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1answer
42 views

Showing that $\bigg\| x - \sum_{k=1}^n \lambda_ke_k\bigg\| \geq \bigg\|x-\sum_{k=1}^n\langle x,e_k\rangle e_k \bigg\|$

Exercise : Let $\{e_1,e_2,\dots, e_n\}$ be an orthonormal set over the Hilbert space $H$. Show that : $$\bigg\| x - \sum_{k=1}^n \lambda_ke_k\bigg\| \geq \bigg\|x-\sum_{k=1}^n\langle x,e_k\...
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4answers
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Showing that $M + N$ is a closed subspace of the Hilbert $H$

Exercise : Let $M, N$ be closed subspaces of the Hilbert space $H$ while it is also assumed that $M \bot N$. Show that the set : $$M + N = \{x+y : x \in M, y \in N\}$$ is also a closed subspace ...
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0answers
34 views

Lemma 5.1.5 from Garth Dales, Introduction to Banach algebra

The problem is from Garth Dales, Introduction to Banach algebra, chapter 5 and Lemma 5.1.5 Lemma: Let $(A, \|.\|)$ be a unital Banach algebra, let $a\in A$ and let $\epsilon>0.$ then there is a ...
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1answer
24 views

Bounded Linear Functional and Riesz's Lemma

Let $ X = C[0,1] $ with the supremum norm $ || \cdot ||_\infty $. Consider the functional $ \phi $ defined for $ f \in X $ by $$ \phi(f) = \int_0^\frac{1}{2} f(x) dx - \int_\frac{1}{2}^1 f(x) dx .$$ ...
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3answers
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Prove that $\Vert\cdot \Vert^2:X\to \Bbb{R},$ where $X$ is a vector space, is convex

Let $X$ be a vector space. I was able to prove that $\Vert\cdot \Vert:X\to \Bbb{R},$ is a convex function, i.e., for all $x,y\in X$ and $\lambda \in [0,1],$ \begin{align} \Vert \lambda x+(1-\lambda)y ...
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1answer
19 views

If $T:C[0,1] \rightarrow \mathbb{R}$ is defined by $T_{x_0}(f)=f(x_0)$ then $||T||=1$.

Let $x_0 \in [0,1]$. Define $T_{x_0}:C[0,1] \rightarrow \mathbb{R}$ by $T_{x_0}(f)=f(x_0)$. $||T||:= \sup\{|T_{x_0}(f)|: ||f||_{\infty} \leq 1\}$ where $||f||_{\infty}:=\max\{|f(x)|:x \in [0,1]\}$. ...
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2answers
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linear isometric embedding from $(\mathbb{R}^2, \| \|_2)$ to $(l^1, \| \|_1)$

I would like to prove the following : There isn't a linear isometric embedding from $(\mathbb{R}^2, \| \cdot \|_2)$ to $(l^1, \| \cdot \|_1)$ I don't know how to prove this. So far I am able to ...
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0answers
19 views

A property of normed vector spaces equipped with a preorder: inequality between positive vectors implies inequality between their norms

Let $V$ be a normed vector space equipped with a preorder $\preceq$. Is there a name for the following property: $$ \forall s, t \in S.\ \mathbf{0} \preceq s \preceq t \implies \|s\| \leq \|t\|\tag{*} ...
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Reason for the term “smooth”

A normed space $X$ is said to be smooth if for $x \in X$ with $||x||=1$ there exists a unique bounded linear functional $f$ such that $||f||=1$ and $f(x)=||x||$. Why the term "smooth" comes?
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Is $C^1([a,b], \mathbb{R}^n)$ a reflexive Banach space?

I want to prove or disprove that $C^1([a,b], \mathbb{R}^n)$ equipped with the norm $||x||=\underset{t\in[a,b]}{\sup}|x(t)|_{\mathbb{R}^n}+\underset{t\in[a,b]}{\sup}|\dot{x}(t)|_{\mathbb{R}^n}$ is a ...
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0answers
26 views

Fréchet derivative: dependency of the choice of norm

Let $ f_1: L^2(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $, $ f_2: H^1(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $ and $f_2: H^1(\Omega)\...
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Adherence of a subset - Intersection of kernel

Having Y, a subspace of X. How can we show that the adherence of Y can be expressed as : Adherence of Y = intersection of { Ker(f) | f element of X* (dual space) , Y contained in Ker(f)} I guess I ...
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1answer
23 views

Convergence in a normed vector space - Linear operator [closed]

Having $X$ a normed vector space. If $f$ is a linear operator from $X$ to $ℝ$ and is not continuous in $0$ (element of $X$) , how can we show that there exists a sequence $x_n$ that converges to $0$ ...
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2answers
57 views

$\operatorname{Ker}(B) \subset \operatorname{Ker(A)}$ if and only if ${\left\| {Ax} \right\|_X} \leqslant \alpha {\left\| {Bx} \right\|_X}$

Is this statement is correct: Let $X$ be a Banach space, and let $A$ and $B$ two continuous operators on $X$ , do we have for some constant $\alpha$ the following $${\left\| {Ax} \right\|_X} \...
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1answer
22 views

Dual space and kernel

Having X , a normed vector space and X* = B(X,R) its dual space ( R is the real numbers). Show that for all f contained in X*, we have that Ker f included in X is a closed subspace. Knowing that X* ...
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1answer
35 views

Matries inequality with norms

Let $P$ and $C \neq0$ a $q \times q$ matrices. I want to prove that there exists a positive constants $\alpha$ such under some assumptions under $P$ we have the inequality $${\left\| {P\left( {I - C} ...
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2answers
30 views

Showing that $\|x \| = \sup_{y \neq 0} \frac{|\langle x,y \rangle|}{\|y\|}$

Exercise : Let $H$ be an inner product space and $x \in H$. Show that : $$\|x \| = \sup_{y \neq 0} \frac{|\langle x,y \rangle|}{\|y\|}$$ Attempt : If $x=0$ then the equality follows imidiatelly ...
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1answer
22 views

Set of nowhere differentiable function in C[(0,1]) is dense

How can we prove with Baire's Theorem that in C[(0,1]), the set of nowhere differentiable function is dense.
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1answer
24 views

Properties of bounded linear operators between two normed spaces

Let $T$ be a linear operator between two normed spaces $X$ and $Y$. Show that a) If $T$ is not bounded then for $\varepsilon>0$ , $\sup \Vert Tx\Vert = \infty$ where $\Vert x \Vert < \...
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1answer
41 views

Show $T(x) := x+sf(x)$ is a bijection with $f$ Lipschitz and $\vert s \vert \lt \frac{1}{L}$

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a function and $L \gt 0$ such that $\Vert f(x)-f(y)\Vert \leq L \Vert x-y\Vert\ $for all $x,y \in \mathbb{R}^n$ Show that $T(x) := x+sf(x)$ defines ...
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1answer
35 views

Banach space with subset whose elements are at least $d\gt 0$ far from each other is not separable

Let $X$ be a Banach space, and $A\subseteq X$ subgroup, where $A$ is not countable, and there is some $d \gt 0$ such that for all $x,y \in A$: $||x-y||>d$. Prove that $X$ is not separable. My ...
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1answer
43 views

Equivalent operator norm

Let $E$ be a Banach space and let $L:E \to E$ be a bounded operator. I want to know when de we have the equivalence between the norms ${\left\| {Lx} \right\|_E}$ and ${\left\| {x} \right\|_E}$. More ...
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1answer
24 views

convex cone in complex Banach space

A convex cone is defined as (by Wikipedia): A convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients. In my research work, ...
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1answer
25 views

A closed convex set in a complex Banach space.

Let $X$ be a Banach space over the field of complex numbers. Let $K$ be a subset in $X$ which contains all linear combinations $Z_1x+Z_2y$ for all $Z_1,Z_2\in \mathbb C$ with $Re(Z_1)\geq0$ & $Re(...
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1answer
15 views

Giving a proof to the weak compactness of the unity ball in a reflexive normed space

So, I am trying to prove that if $X$ is a normed and reflexive space, its unity ball, that is $B$=$\lbrace x\in X : \parallel x\parallel =1\rbrace$, is weakly compact. For that matter, I have already ...
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1answer
22 views

On the closure of the convex hull of a sequence in normed spaces

Let $E$ be a infinite dimensional normed spaces and $(x_n)_{n=1}^\infty$ be a sequence in $E$ converging to zero. Is it true that the closure of the convex hull of the set $\{x_n: \ n\in \mathbb{N}\}\...
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5answers
77 views

If there are $2$ linearly independent vectors $x,y \in X$ such that $||x+y||=||x||+||y||$, then the unit sphere $S(X)$ contains an interval

Let $S(X)= \{x \in X: ||x||=1\}$ be the unit sphere in $X$. Assume that there are $x,y\in X$ linearly independent such that $||x+y||=||x||+||y||$. Prove that $S(X)$ contains the following set:$[x,y]=\{...
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1answer
26 views

Open unit ball in integral norm is open in supremum norm

I have to show the following I know I have to find an $\epsilon$ such that $B_\infty(f,\epsilon)$ is in $B_1(0,1) \forall f$ . I just cannot figure out what this epsilon could be. Also I don't know ...
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1answer
31 views

Show “identity” with parallelogram law in a vectorspace

So I have a vectorspace with the norm $||\cdot||$. And I know the parallelogram law is valid because $(V, <\cdot , \cdot >)$ is a Hilbert space. So I know: $$ ||x+y||^2+ ||x-y||^2=2(||x||^2+||y||...
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1answer
31 views

Are $\lVert .\rVert _2$ and $\lVert . \rVert_{F} $ equivalent for a matrix?

If $A\in mat_{n\times n}(\mathbb C) $ and $\lVert A\rVert _F $ is equal to the Frobenius norm of $A$ and $\lVert A\rVert_2=(\lambda_{\max }(A^*A))^{\frac{1}{2}} $, Are these norms equivalent? I know ...
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1answer
7 views

Coercive bilinear form for maximum norm

Let $f$ be a differentiable function. Denote a bilinear form by $$b(f,f) = \int_{0}^{1} \bigg( \frac{d f(x)}{dx} \bigg)^{2} dx.$$ Given $f(0) = 0,$ we want to show that $$a \cdot b(f,f) \geq ||f||_{\...
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1answer
25 views

Balls in finite dimensional normed spaces

Let $X$, $Y$ be normed linear spaces and $Y$ be finite dimensional. Suppose $T \in B(X,Y)$ such that $T$ is surjective. Prove that there is some $\delta > 0$ so that $B_\delta(0_Y ) \subseteq T(B_1(...
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1answer
18 views

Finding operator norm of functional

I am given a functional $F:C([0,1],\|x\|_\infty) \to \mathbb C $ by formula $$F(f)=2\int_0^{1/2}f(t)dt \text{ - } \int_{1/2}^1f(t)dt$$ I should find its operator norm. I've found that $\|F\| \le 3/2$...
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2answers
35 views

Minimizing norm of inner product space

Let $X$ be an inner product space and $\{x_1,....,x_{n}\}$ be orthonormal prove $\bigg|\bigg|\sum_{k=1}^{n} x-c_kx_k\bigg|\bigg|$ is minimized by $c_n=(x,x_{n})$ Thoughts since $x_k$'s are ...
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1answer
31 views

There exists a unique extension $\hat{T}$ of a bounded linear operator $T$.

I am trying to prove the following theorem : Theorem : Let $X,Z$ be Banach (normed) spaces and $Y$ be a dense subspace of $X$. Let $T:Y \to Z$ be a bounded linear operator. Then, there exists a ...