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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43 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
51 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 ...
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32 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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123 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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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 ...
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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 ...
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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 ...
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2answers
53 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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89 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
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 ...
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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
30 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 ...
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157 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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25 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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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. ...
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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 ...
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3answers
83 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
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 ...
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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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68 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 $A=\{x\in E \;|\; \forall t\in [0,1], (1-t)a+tx\in U\}$ ...
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96 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 ? ...
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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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14 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
56 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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76 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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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
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3answers
184 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 $\|\dot x\|^2\to a$, ...
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1answer
56 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 ...
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1answer
32 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 ...
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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 ...
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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 ...
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1answer
40 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 ...
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0answers
29 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
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1answer
66 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 ...
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2answers
44 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 ...
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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 ...
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0answers
46 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 ...
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17 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
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31 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 ...
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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 ...
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100 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
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22 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 \ ...
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1answer
46 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 ...
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1answer
46 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 ...
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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
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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
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0answers
30 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 ...
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2answers
35 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
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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 ...