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
0answers
29 views

$d(x,L)=\max\{f(x) \,| \, f\in L^{\perp},\, \|f\|=1\}$

Let $X$ be a normed space and $L$ its subspace. Let $L^{\perp}$ be a set of all functional of whose kernel contains $L$. Then $d(x_0,L)=\max\{f(x_0) \,| \, f\in L^{\perp},\, \|f\|=1\}$ I read a ...
1
vote
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
vote
0answers
27 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
votes
1answer
61 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 ...
1
vote
1answer
61 views

Linear and nonlinear operator on normed space and its properties

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 equivalent in linear ...
0
votes
1answer
23 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 ...
1
vote
1answer
27 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: ...
0
votes
0answers
37 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: $\lim\limits_{n\to\infty}x_n{=}0_X$ ...
-1
votes
0answers
16 views

When is a norm of identity one?

Is there a specific condition that makes a norm (any norm) of identity equal to one in any Banach spaces? Thanks.
1
vote
0answers
28 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 ...
1
vote
1answer
24 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.
0
votes
1answer
19 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
votes
1answer
38 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
votes
3answers
76 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 ...
15
votes
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, ...
0
votes
1answer
29 views

What does this assumption imply in case $X$ is a complex inner product space?

If $X$ is a real inner product space and $x,y\in X$ satisfy $\|x\|=\|y\|$, then $(x-y)\perp (x+y)$. What does this assumption imply in case $X$ is a complex inner product space? My Work: I proved ...
0
votes
1answer
35 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 ...
1
vote
1answer
40 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 ...
0
votes
1answer
58 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
votes
1answer
45 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 ...
1
vote
1answer
42 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 ...
1
vote
1answer
48 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
votes
1answer
29 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 ...
4
votes
2answers
114 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 ...
1
vote
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
votes
1answer
20 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 ...
0
votes
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 ...
3
votes
2answers
51 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: $\lim\limits_{n\to\infty}x_n{=}0_X$ $\Longrightarrow$ ...
1
vote
0answers
86 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 $$ ...
1
vote
1answer
17 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 ...
1
vote
1answer
34 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 ...
1
vote
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 ...
0
votes
1answer
29 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
152 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. ...
0
votes
0answers
23 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 ...
0
votes
0answers
15 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. ...
0
votes
1answer
35 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 ...
4
votes
3answers
75 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 ...
0
votes
1answer
17 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 ...
1
vote
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
2
votes
1answer
67 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
votes
1answer
86 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
votes
0answers
36 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 ...
0
votes
0answers
13 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 ...
4
votes
2answers
52 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?
6
votes
0answers
72 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.
0
votes
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
179 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
55 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
31 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 ...