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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28 views

Is the countable product of normed spaces a normed space with this “norm”

Consider the product $\mathcal{N} = \mathcal{N}_{1} \times \mathcal{N}_{2} \times \mathcal{N}_{3} \times \dots$ where $\mathcal{N}_{1}, \mathcal{N}_{2}, \mathcal{N}_{3}, \dots$ are normed spaces with ...
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41 views

Showing that if any non-zero $f \in X^*$ takes its maximum value on the unit sphere at most once, then X is strictly convex

Let $(X, \| \|)$ be a normed space. I'm trying to show that if all non-zero $f \in X^*$ take their maximum value on the unit sphere at most once, i.e. $\forall f \in X^* - \{0\}$ there is at most one ...
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1answer
27 views

A question about the natural identification between $Y$ and $Y^{**}$, with $Y$ normed space. Is the following fact obvious?

Let $X$ be a reflexive normed space, $Y$ a normed space and $T: X \to Y$ a linear operator. I consider a sequence $\{x_n\}$ in X and the sequence $\{T(x_n)\}$ in $Y$. I know that there is a ...
1
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0answers
29 views

the second derivative for a composite function

Let $f: U\rightarrow \mathbb{R}$ where $U\subseteq X$ a normed vector space and suppose $f(\mathbf{u})\in C^{2}$.Let $\mathbf{x}\in U$ and let $r>0$ be such that $B(\mathbf{x},r)\subseteq U. $ ...
2
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1answer
65 views

Solutions to $u \circ v - v \circ u = \mathrm{Id}$

Let $(E,\Vert \cdot \Vert)$ be an infinite-dimensional normed vector space and $\mathcal{L}_{c}(E)$ denote the ring of continuous endomorphisms of $E$. I would like to determine whether the equation ...
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1answer
88 views

Linear and nonlinear operator on normed space and its properties

My first question is : We know every bounded linear operator is continuous operator , and every continuous linear operator is bounded operator , in other words the continuousness and boundedness are ...
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1answer
42 views

Prob. 3, Sec. 2.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $C[-1,1]$ denote the normed space of all (real or complex-valued) functions defined and continuous on the closed interval $[-1,1]$ on the real line, with the norm given by $$\Vert x ...
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1answer
44 views

$H_\infty$ norm of a transfer function by Matlab

Consider the simplest case: $$H(z) = \frac{z+1}{z+2}$$ I use two methods to find $H_\infty$-norm: ...
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40 views

Stuck because a possible error in the statement of a functional analysis exercise.

I was given this exercise: Let $X$ and $Y$ be two normed spaces, with $X$ a reflexive space. Suppose that $T:X\rightarrow Y$ is an operator such that: $\lim\limits_{n\to\infty}x_n{=}0_X$ ...
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0answers
31 views

Is this fact true $T:X \longrightarrow Y$ linear and bounded operator with $X$ reflexive then $T$ is compact?

I wonder if this fact is true. I consider a linear operator $T:X \longrightarrow Y$, with $Y$ and $X$ two normed spaces. I suppose that $X$ is a reflexive space and that $T$ a bounded operator. Is ...
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1answer
30 views

Nonlinear discontinuous convex function

Let $X$ be a normed vector space. Construct nonlinear discontinuous convex function $f:X \rightarrow \mathbb{R}$. I tried something with $\frac{-1}{\|x\|}$ but had no success.
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1answer
23 views

Intersection of a Hyperplane and a Subspace

Let $Y$ be a dense linear subspace of a normed space $X$, and let $M$ be a closed hyperplane in $X$. I'm trying to show that $M \cap Y$ is a hyperplane in $Y$ and dense in $M$. I've been trying to ...
2
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1answer
41 views

Banach space of p-Lipschitz functions

Given $p\in\mathbb{R}$, consider the space: $$ Lip(p) = \left\{f:[0,1] \longrightarrow \mathbb{R} : \mbox{ $f$ is $p$-Lipschitz} \right\}$$ i.e.: there is $M>0$ such that ...
2
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3answers
80 views

Infinite dimensional spaces other than functional spaces

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences. I ...
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4answers
1k views

How to develop an intuitive feel for spaces

I'm a physicist who's currently delving deeper into what I would call more 'hardcore' maths (e.g. FEM and control theory). Every now and then, I come across various spaces, such as vector spaces, ...
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1answer
46 views

How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
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1answer
42 views

Why is $A:X\to Y$ linear between two normed spaces is continuous iff bounded?

Why is it that every linear operator $f: \mathbb R^n \to \mathbb R^m$ is bounded and therefore continuous, but why is it that $A:X \to Y$ between two normed spaces is continuous iff bounded? That ...
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1answer
78 views

Prob. 14, Sec. 2.7 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
4
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1answer
49 views

Compute $\| T \|$ with the norm $\| \cdot \| := \max_{j=1,\ldots,n} (|\cdot_j|)$

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the normed space of all ordered $n$-tuples of real numbers with the norm defined as ...
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1answer
45 views

Upper bound for norm of matrix (cf. Example 2.7-7 in Erwine Kreyszig's book)

Let $A \colon = [\alpha_{ij}]_{m\times n}$ be a given $m \times n$ matrix of real numbers. Let $\mathbb{R}^n$ be the norm space of all ordered $n$-tuples of real numbers with the norm defined as ...
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1answer
57 views

Holder's inequality with expectation norm

One defines the ``p-expectation norm" of a function $f$ as $\vert f \vert_p = (\mathbb E ( f^p) )^{ \frac{1}{p}}$. What is the intuition for this norm? Now why are the following true? $\vert ...
2
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1answer
37 views

Show that dual space of $R^n$ with norm 3 is equal to the $R^n$ with norm 1.5.

How can one prove that dual space ($R^n$,$||.||_3$)*= ($R^n$,||.||1.5). How to go about using the holder's inequality? Any help will be appreciated! Hint: I know I've to use holder inequality to make ...
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2answers
126 views

What is the norm of the pre-multiplication by a fixed matrix operator?

Let $A \colon= \left(\alpha_{ij} \right)_{m\times n}$ be a given $m \times n$ matrix of complex numbers, and let the operator $T \colon \mathbb{C}^n \to \mathbb{C}^m$ be defined by $$T(x) \colon= Ax ...
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0answers
19 views

Let $V$ be a NLS (over $\mathbb R$ ) of dimension $>1$, then for any $0 \le a < b$ , is the set $\{v \in V : a < ||v|| < b\}$ connected?

Let $V$ be a normed linear space (over $\mathbb R$ ) , then for any $0 \le a < b$ , is the set $\{v \in V : a < ||v|| < b\}$ connected ? I know that if $V$ is the space of complex numbers ...
0
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1answer
23 views

show that the normed linear vector space $\mathbb{R}^n$ with norm $||x|| = \max{(|x_1|, |x_2|, … , |x_n|})$ is complete.

I am trying to show that the normed linear vector space $\mathbb{R}^n$ with norm $||x|| = \max{(|x_1|, |x_2|, ... , |x_n|})$ is complete. My approach was as follows: First, construct a Cauchy ...
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1answer
30 views

Show that a linear form $\mathbb{R}^n \to\mathbb{R}$ is continuous

$f(x)$=$n∑k=1$ $g$($x_k$) ou $x_k$ is the kth component of the vector x. $x_k=\langle e_k,x\rangle$. I have the option of showing this with sequences (which I dont know how, I never understood how to ...
2
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2answers
55 views

$T$ linear operator s.t. $\lim\limits_{n\to\infty}x_n{=}0_X$ $\Longrightarrow$ $\lim\limits_{n\to\infty}T(x_n){=}0_Y$ then $T$ is bounded

Let $X$ and $Y$ be two normed spaces, with $X$ a reflexive space. I suppone that $T:X\rightarrow Y$ is an operator such that: $\lim\limits_{n\to\infty}x_n{=}0_X$ $\Longrightarrow$ ...
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0answers
95 views

Properties maintained by the direct sum of normed spaces.

Let $(X_i)_{i=1}^\infty$ be a sequence of normed spaces. We define the $\ell_p$-direct sum $[\bigoplus_{i=1}^n X_i]_p$ as the normed space of elements $(x_i)_{i=1}^n\in \prod_{i=1}^n X_i$ with norm $$ ...
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1answer
19 views

Prove that $S:F \to \mathcal{L}(E,F)$ is a topological isomorphism

Let $E$ and $F$ be normed spaces, $E \neq \{ 0 \}$ and $x_0 \in E \backslash \{ 0 \}$, $x_0 \in E'$ such that $x_0'(x_0)=1$. Prove that the function $S: F \to \mathcal{L}(E,F)$, $S(y)=T_y$ defined ...
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1answer
36 views

How can I fix this proof using transfinite induction of the existence of bases of normed vector spaces?

I want to prove that every normed vector space has a basis. The following proof relies on the principle of transfinite induction. I believe that it is flawed because I'm not so sure if it's possible ...
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0answers
28 views

Is there a proper subspace of $T$ that includes $T^{n}x$ for all $n\in \mathbb{N}$?

Suppose $E$ is a normed space, $T$ is a bounded operator from $E$ to $E$ and $B_E$ is closed unit ball of $E$. If there is $\exists \epsilon >0$ and $% \exists y\in B_{E}$ such that $\left\Vert ...
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1answer
34 views

Integrate function under $L^2$ Norm

I am following the book of Salsa (2008), BTW very good book, and I found the this example that I can't really understand how he expanded the integral. Let's say you want to use the $L^2$ norm under a ...
6
votes
1answer
162 views

Converse of uniform boundedness principle

The uniform boundedness principle says if we have a collection of bounded linear operators $\Gamma$ from a banach space $X$ into a normed vector space $Y$, which is pointwise bounded on $X$, i.e. ...
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31 views

Why is this an nuclear norm?

The following figure is from a paper: "The Convex Geometry of Linear Inverse Problems" Just on p.3. As in the figure, the red lines are $2 \times 2$ symmetric unit-Euclidean-norm rank-one ...
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0answers
17 views

The meaning of equivalence in norm.

Ask a elementary question: In WIKI: " However all these norms are equivalent in the sense that they all define the same topology." I think "these norms" here mean $l_1, l_2,...l_{inf}$ norms. ...
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1answer
36 views

Functional analysis, normed space

Can you please help me with this exercise.. I have to check if in interval $[a,b]$ continuously differentiable functions $x=x(t)$ norm can be defined as: $$\vert x(b) - x(a) \vert + \max_{a \leq t ...
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3answers
91 views

Can one find a stronger norm on a Banach space?

Given a Banach space $V$ of infinite dimension with norm $\|\cdot\|_1$, is that possible to find a norm $\|\cdot\|_2$ on $V$ such that the topology induced by $\|\cdot\|_2$ is strictly stronger than ...
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1answer
18 views

plane angle calculation problem

in calculating the angle between the plane $2x + y -2z +4 = 0$ and $z$ axis I got that the angle between the normal and $z$ axis is $131.81$. however if I take $90°$ minus that I get a negative angle ...
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2answers
32 views

How can I prove this operator is not continuos

Let $X$ the normed space of all polynomials on $J=[0,1]$ such that $||x||$=max$|x(t)|$ $t \in[0,1]$ and we have the following operator $Tx(t)=x'(t)$ prove this operator is not continuous
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1answer
70 views

Proving a set is open.

Let $E,||\cdot||$ be a finite dimensional normed space over $\mathbb R$. Let $U$ be an open subset of $E$ and $a\in U$ Let $A=\{x\in E \;|\; \forall t\in [0,1], (1-t)a+tx\in U\}$ ...
5
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1answer
101 views

When does continuous (or uniformly continuous ) function between normed linear spaces carries bounded sets to bounded sets ?

I know that if $f:\mathbb R^m \to \mathbb R^n$ is continuous then $f$ carries bounded sets to bounded sets . What if we say $X,Y$ are normed linear spaces and $f:X \to Y$ where $f$ is continuous ? ...
3
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0answers
38 views

Find all $ p \ge 1 $ for which the Hölder norm $\|\cdot\|_p $ is generated by a scalar product.

Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product. We know that norm is generated by a scalar product ...
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0answers
16 views

Mahalanobis distance to the ellipsoid center

I am confused about the following description: If we parameterize the ellipsoid $E$ as: $E = \{x|\ ||Ax-b||_2 \leq 1\}$. $A \in S_{++}^n$ Then the Mahalanobis distance to the ellipsoid center is ...
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2answers
57 views

Convergence Problem in Normed Space

Probably easy, but I'm stuck atm: A sequence converges in norm 1 if and only if it converges in norm 2, for all sequences. Are the two norms necessarily equivalent?
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0answers
77 views

One more AC equivalence question

Is "Every vector space admits a norm" weaker than AC? I know that the statement follows from "Every vector space has a basis", which is equivalent to AC.
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0answers
19 views

Convergence of a sequence in a normed vector space [duplicate]

help with homework problem... I feel like its easy, I'm just missing something Show that $\{||x_k||\}$ converges in $\mathbb{R}$ if $\{x_k\}$ converges in a normed vector space V. merci :) its ...
4
votes
3answers
193 views

If $\|u\|^2 = c^2$, then $\|\dot x\|^2 \to a $ for $\ddot x = -\dot x + u(t)$?

I am dealing with the next simple equation $$ \ddot x = -\dot x + u(t), $$ where $u, x\in\mathbb{R}^m$, with $m \geq 1$, and I am wondering if for $\|u\|^2 = c^2 > 0$ then $\|\dot x\|^2\to a$, ...
0
votes
1answer
60 views

Completion of a vector space inside a given Banach space

Let $X$ be a normed vector space and $Y$ be a Banach space such that $X$ is continuously embedded into $Y$ (this will be denoted by $X\hookrightarrow Y$ in the sequel). Is it always possible to find ...
3
votes
1answer
36 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
0
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1answer
22 views

Dimension of quotient normed linear space

Suppose $M$ is a normed linear space. $L$ and $N$ are two closed subspaces of $M$ such that $L \subseteq N$. Then $L$ is a closed subspace of $N$. Let $\text{dim}(M/N)=r$ and $\text{dim}(N/L)=s$. My ...