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

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
28 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
20 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
71 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
64 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
83 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
34 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
51 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
175 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
54 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 ...
2
votes
0answers
23 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
votes
1answer
21 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 ...
2
votes
1answer
46 views

Show that $X$ is Banach space and describe $X^*$.

Let $X=L^2(\mu)\times L^2(\mu)=\{(f,g)|f,g\in L^2(\mu)\}$ be the linear space normed by $\|(f,g)\|=(\|f\|_2^3+\|g\|_2^3)^{1/3}$. Show that $X$ is Banach space and describe $X^*$. My Work: We ...
2
votes
1answer
37 views

Show that given $ϵ>0,$ there exists $x∈X$ such that $∥x∥=1$ and $d(x,M)>1−ϵ.$

Let $X$ be a normed linear space and $M$ be a proper closed linear subspace of $X$. Show that given $ϵ>0,$ there exists $x∈X$ such that $∥x∥=1$ and $d(x,M)>1−ϵ.$ My Work: Let $ ϵ>0$. Since ...
2
votes
0answers
27 views

Exercise in Hahn-Banach Theorem; Finding linear functional $-p(-x)\leq f(x)\leq p(x)$

(The following exercises are in Kreyszig's book 218 page; EXE 10) I want to solve the following exercise : If $X=l^\infty$, let $p(x)=\lim\sup x_i $, whichi is sublinear. Then find a linear functional ...
2
votes
1answer
63 views

About a technique used in the proof of Hahn-Banach Theorem

Recall Hahn-Banach (cf. Kreyszig's book) : If $X$ is a real vector space with a sublinear functional $p$ and if $f$ is linear on a subspace $Z$ with $p(z)\geq f(z),\ z\in Z$, then there exists an ...
2
votes
2answers
42 views

prove that $\{Lx:\|x\|\leq 1 \}=\mathbb{C}$

Let $X$ be a linear normed space over $\mathbb{C}$. If a linear functional $L$ on $X$ is not continuous, prove that $\{Lx:\|x\|\leq 1 \}=\mathbb{C}$ Clearly $\{Lx:\|x\|\leq 1 \}\subseteq ...
0
votes
1answer
32 views

How does finite linear combinations of the $x_n$'s looks like?

Let $X$ be a normed linear space and let $\{x_n\}\subseteq X$. Prove that $x\in X$ is the limit of finite linear combinations of the $x_n$'s iff $Lx=0$ for all continuous linear functionals $L$ on ...
2
votes
0answers
38 views

Question about Stone-Weierstrass theorem

I have a question about Stone - Weierstrass theorem. In the space $C[0,2\pi]$ of continuous functions on $[0,2\pi]$ with the sup norm. Consider the spaces $M$ of all trigonometric polynomials. It's ...
0
votes
0answers
16 views

A question involving normed spaces and strictly convex spaces

Let $(X, \| \cdot \|_X)$ be a normed space and let $\| \cdot \|$ be a norm on $X$ such that $(X, \| \cdot \|)$ is strictly convex. How can I find a strictly convex space $(Y, \| \cdot \|_Y)$ and a ...
2
votes
0answers
27 views

Equivalence of norms between two Banach spaces

I just learnt open mapping theorem. And I met a statement online asserting that If $X$,$Y$ are Banach space, and $T:X\to Y$ be a continuous bijection, then norms for $X,Y$ are equivalent. Can we ...
1
vote
1answer
26 views

A question involving norms

Let $(X, \| \cdot \|_X), (Y, \| \cdot \|_Y)$ be normed spaces and $T : X \rightarrow Y$ a bounded operator. Let $x, y \in X$ and let the norm on $X$ $$ \|x\| = \|x\|_X + \|Tx\|_Y. $$ I can't show that ...
1
vote
2answers
93 views

What are the best books for studying functional analysis in the world

I want to ask you maybe strange question but I really need answer What are the best books for studying functional analysis After Afew week I start study in master so I want references
0
votes
1answer
20 views

$f \in \mathcal{C}_{0}(X) \Rightarrow \sup_{x \in X} |f(x)| = \max_{x \in X} |f(x)|$

For a topological space $X$ we define $$\mathcal{C}_{0}(X) : = \left\lbrace f \colon X \longrightarrow \mathbb{C} \ \text{continuous} \colon \forall \, \varepsilon >0 \ \exists \, K \subseteq X \ ...
0
votes
1answer
45 views

Why if $T$ is not a bounded operator then exists $ (x_n) $ that converges to $ 0_{X} $ for which $ \| T(x_n) \| \geq n^2 $ for all $ n $?

Let $X$ and $Y$ be normed spaces. Suppose that $ T: X \to Y $ is a linear operator and assume that $T$ is not bounded. Why with these assumptions can I say that exists a sequence $ (x_{n})_{n \in ...
1
vote
1answer
45 views

$\|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}}$

I want to prove that $$ \|f\|_2\le\|f\|_4^{\frac{2}{3}}\|f\|_1^{\frac{1}{3}} $$ I proved it by Holder inequality. But this is an exercise under "Interpolation". So I guess it can be proved using ...
0
votes
0answers
22 views

Meaning of Normed Space

I have the following notations: 1.$(L^1\bigcap L^\infty)(0,A)$. 2.$L^\infty ((0,A)^2)$ 3.$L^\infty(Q)$ where Q =$(0,T) \times (0,A)$. Can someone explain to me what does the L norm represents, their ...
-1
votes
1answer
62 views

Study the convergence of the following sequence [closed]

How can I study the convergence for the following sequences in $L^2$ space ? $x_n=\big(\frac{1}{n},\frac{1}{n},\frac{1}{n},\frac{1}{n},..,\frac{1}{n},0,0,0,...\big)$ $n^2$
1
vote
2answers
23 views

For $X \in M(n,\mathbb R)$ , let $||X||:=\sqrt{Trace(AA^t)}$ , then $||AB|| \le ||A||\space||B|| , \forall A,B \in M(n,\mathbb R)$?

Let $M(n,\mathbb R)$ be the set of all square matrices of size $n$ with real entries . For $A \in M(n,\mathbb R)$ , let $||A||:=\sqrt{Trace(AA^t)}$ , then is it true that $A,B \in M(n.\mathbb R) ...
0
votes
0answers
26 views

Proving finite dimensional normed linear space is complete , without using equivalence of norms on finite dimensional vector spaces

Every finite dimensional normed linear space , over the field of real numbers , is complete . I know a proof of this result by using "every norm on a finite dimensional real vector space is equivalent ...
1
vote
2answers
33 views

Is the parallelogram equality satisfied in $l^1$?

I can't show that the parallelogram equality is satisfied / or is not satisfied in $l^1$. If $(v_n), (w_n) \in l^1$, then we have $$ || (v_n) + (w_n) ||_1^2 + || (v_n) - (w_n) ||_1^2 = || (v_n + ...
2
votes
1answer
39 views

Determining whats the Induced Metric

I have the normed space $({\rm Lip}([0,1]), \|\cdot\|)$, where ${\rm Lip}([0,1])$ is all Lipschitz functions from $[0,1]$ to $\Bbb R$, and $$\|f\|=|f(0)|+\sup_{0\le x,y\le ...
12
votes
1answer
176 views

In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?

$(V,\|\cdot|)$ be a real normed linear space such that $\|x\|=\|y\|$ implies $\lim\limits_{n\to\infty} \|x+ny\|-\|nx+y\|=0$, then is it true that the norm comes from an inner-product space ?
1
vote
1answer
32 views

A real vector space is an inner product space if every two dimensional subspace is an inner product space ?

Is it true that a vector space over the field of real numbers is an inner product space if every two dimensional subspace is an inner product space ? does it have anything to do with Neuman-Jordan's ...
0
votes
0answers
15 views

Looking for a simpler proof of Day's characterization of inner-product spaces and related things

I know the theorem that if $(V,||.||)$ is a real normed linear space such that the parallelogram identity $||x+y||^2+||x-y||^2=2(||x||^2+|y||^2)$ holds , then the norm comes from an inner-product ...
0
votes
1answer
40 views

Showing a Set is Dense in $C[a,b]$

Show that $C_0[a,b]:= \{f \in C[a,b] : f(a)=f(b)=0 \}$ is $\| \cdot \|_2$-dense in $C[a,b]$. Is it also $\| \cdot \|_{\infty}$-dense in $C[a,b]$? In the relevant chapter of my textbook, I am given ...
0
votes
2answers
21 views

$f,g \in [0,1] , f<g $ , when is $U:=\{h \in C[0,1]:f(t)<h(t)<g(t), \forall t \in [0,1] \}$ a ball in $C[0,1]$ with respect to the sup metric

Let $f,g:[0,1] \to \mathbb R$ be continuous functions such that $f(t)<g(t),\forall t \in [0,1]$ , then under what additional conditions on $f,g$ can we conclude that $U:=\{h \in ...
0
votes
2answers
13 views

A question on Matrix limits

Consider $M(n, \mathbb R)$ , the set of all real square matrices of size $n$ , as an NLS with norm $||A||:=\sqrt{\sum_{i=1}^n \sum_{j=1}^n a_{ij}^2}=Trace(AA^t)$ , then is it true that a sequence of ...
2
votes
1answer
34 views

A consequence of the open mapping theorem

We let $f$ be a bounded and surjective linear map from the Banach space $X$ onto the Banach space $Y$ and put $$ r_0=\inf\{r: f(B^X(0,r)\supset B^Y(0,1)\}. $$ Using the open mapping theorem, I have ...
1
vote
0answers
28 views

Equivalence of norms in $\mathbb R^n$

Does anyone know some proofs where we use fact of equivalence of norms(I think some L continuity use this fact). Any literature of proofs would be very useful. Thank's a lot.
2
votes
1answer
33 views

Multiplication Operator and Supremum Norm

Let $m\in C[a,b]$. Consider on $(C[a,b], \|\cdot \|_{\infty})$ the multiplication operator $A: C[a,b] \to C[a,b], \quad Af = mf$. Prove that $\|A\| = \|m\|_{\infty}$. In my book, we are given the ...
1
vote
1answer
45 views

Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...