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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17 views

How to show that every Cauchy sequence converges in some normed linear space?

(Hunter and Nachtergaele, 1.6) Using the fact that R is a complete metric space with respect to Euclidean distance, show that Rn is a Banach space when equipped with (a) the Euclidean norm (b) the ...
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0answers
19 views

Completeness of metric, normed and inner product spaces

For a metric space to be complete, it needs to have all cauchy sequences converge in the metric. 1) For a normed space to be complete, does it need Cauchy sequences to converge in the norm or in the ...
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18 views

I need help from you by answering the qoestion below [on hold]

Adama Science and Technology University Department of Mathematics Functional Analysis I(Work sheet II) 1. Let X be a normed space. If x, y  X and ||x + y|| = ||x|| + ||y|| then show that || ax + ...
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1answer
21 views

Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
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1answer
39 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...
2
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2answers
38 views

The norm of a bounded linear operator has this formula: $\|T\| = \sup_{\|v\| = 1} \|T v\|$

Trying to prove $\|T\| = \sup_{\|v\| = 1} \|T v\|$, given $\|T\| := \inf_{C \geq 0} \{C: \|Tv\| \leq C\|v\|\}$. I know that $\|T(v)\| = \|T(\alpha \hat{v})\| \leq C\|\alpha \hat{v}\|$ for $v = ...
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1answer
52 views

Example of Heine-Borel Theorem

For a subset $S$ of Euclidean space $\mathbb R^n$, S is closed and bounded if and only if S is compact (that is, every open cover of $S$ has a finite subcover). I need an example or ...
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1answer
40 views

If $(V,\|\cdot \|)$ is a finite dimensional space, then all norms are equivalent. [duplicate]

I want to show that if $(V,\|\cdot \|)$ is a finite dimensional space, then all norms are equivalent. I have shown that if $\dim V=m$ all norms $$\|x\|_p=\sqrt[p]{x_1^p+...+x_m^p}$$ are equivalent, ...
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3answers
48 views

Equivalence of norms problem.

How would I show that $\|\cdot\|_3$ and $\|\cdot\|_\infty$ are equivalent norms on $\mathbb R^2$? I understand that to say two norms are equivalent, then there exist two real constants, $m,M$ such ...
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2answers
65 views

What does ||u|| mean?

What does $\left\Vert \mathbf{u}\right\Vert$ mean in this equation? How would this equation be performed? I'm extremely terrible in discrete mathematics and a simplistic answer would be ideal. (Don't ...
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1answer
28 views

What can we say about open unit balls of sup-norm and integral-norm

Consider the normed linear spaces $X_1=(C[0,1], ||.||_1)$ and $X_{\infty}=(C[0,1],||.||_{\infty})$ , where $C[0,1]$ denotes the vector space of all continuous real valued functions on $[0,1]$ and ...
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2answers
25 views

Can we relax the triangle inequality for $\| v \|$ = $\|v - v_0 + v_0\|$?

Given some vector $v$ on vector space $X$ with a norm $\| \cdot \|$ Then $\| v \|$ = $\|v - v_0 + v_0\|$ where $v_0$ is some other vector is it legal to then write $\| v - v_0 + v_0 \| = \|v -v_0\| ...
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0answers
27 views

Banach contraction mapping to find unique solution of $y(x) = \int_0^1 G(x,s) f(s,y(s)) ds$

Problem statement: Use the Banach contraction mapping theorem to show that $$y(x) = \int_0^1 G(x,s)\ f(s,y(s))\ ds$$ has a unique, continuous solution $y = y(x)$ if (1) $G(x,s) \ge 0$ for $0 \le x, ...
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2answers
24 views

Intuition of a norm vector space/ infinite dimensional vector space

I'm finding it terribly difficult to build an intuition of what a norm vector space and an infinite dimensional vector space is. There aren't any good notes online that builds the intuition-most ...
2
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2answers
18 views

Matrix tensor indices

Suppose I have an orthonormal basis $$B = \left \{ u_{i} \right \}_{i=1}^{\infty}$$ Then for a matrix $K$, do I represent it as $$K = \sum_{j,k=0}^{\infty}k_{jk}\left ( u_{i}\bigotimes u_{j} ...
2
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1answer
31 views

$f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$ ; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$?

Let $f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$ ? I need a proof if it is true ; or any modification ...
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2answers
26 views

Problem in showing that a norm is a norm on one space, but not on another.

I have the following question from a past paper: "Consider the two sets, $$A=\{g\in C^1([0,1]):g(0)=g(1)=0\}$$ and, $$B=\{g\in C^1([0,1]):g'(0)=g'(1)=0\}$$ both subsets of the vector space ...
3
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2answers
62 views

Find a norm so that its closed unit ball is the area between $y=x^2-1$ and $y=1-x^2$

As the title specifies, I need to find an explicit formula for a norm $|||\cdot|||$ so that: $$B_{|||\cdot|||}=\{\mathbf{x} : ||| \mathbf{x}|||\le1 \}$$ where $\mathbf{x}=(x,y)\in\mathbb R^2$, is ...
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0answers
24 views

Why is convexity of $S$ needed in these questions?? (Bachman &Narici, Q-17,18)

Let $X$ be a Hilbert Space and let $\{S\}$ be a Convex set in $X$. Let $d=\inf_{x \in S}\|x\|$ . Prove that, if $\{x_n\}$ is a sequence of elements in $S$ such that $\lim_n \|x_n\|=d$, then $\{x_n\}$ ...
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33 views

Prove the norm axioms

Prove the norm axioms for example 7 . Thank you. :)
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2answers
62 views

Find the precise conditions under which we have $\|x+y\| = \|x\|+\|y\|$

In $\mathbb{R}^n$ consider (the norm infinity) $\|x\|=\max|x(i)|$ where $1\leq i\leq n$. Find the precise conditions under which we have $\|x+y\|=\|x\|+\|y\|$. Thank you for your helping. :)
2
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2answers
34 views

The norm of the extension of an operator

If $T_0$ is the identity operator from $E$ to $E$, where $E$ is a subspace of $F$ and both of them are Banach spaces (maybe not needed). If the bounded linear operator $T$ is an extension of $T_0$ ...
6
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1answer
32 views

A proper subspace of a normed vector space has empty interior.

In a vector normed space $E$, prove that all vectorial subspace $F\neq E$ has a interior empty. My approach:We consider, the open ball $B\subset F$, with $F$ proper subspace of $E$. If $x\notin F$, ...
4
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3answers
35 views

Completness of Normed spaces.

I want to prove the following proposition If $(X,||\cdot||)$ and $(X,||\cdot||')$ are homeomorphic, then $(X,||\cdot||)$ is complete if and only if $(X,||\cdot||')$ is complete. So, I only know the ...
2
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1answer
51 views

$\ell^p\!,$ for $p\neq2$, is not an inner product space. [duplicate]

Consider sequence spaces of the form, $$\ell^p=\Big\{x=\left(x_j\right)_{j=1}^\infty \mathrel{}\big|\mathrel{} \sum_{j=1}^\infty \left\lvert x_j \right\rvert ^p\lt\infty\Big\}$$ for $1\le ...
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1answer
21 views

Point about the theorem and proof of the inner product being a continuous function.

In moving to show that the inner product, $\langle\cdot,\cdot\rangle$ is a continuous function I have the following theorem in my notes (also on page 59 of "Linear Functional Analysis", Rynne and ...
2
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1answer
19 views

Can absolute scalability be 'relaxed' to an equivalent condition in the properties of a norm?

All norms on a vector space $V$ must satisfy for any $x\in V$ $$\Vert \alpha x \Vert = \vert \alpha \vert \Vert x \Vert $$ for any scalar $\alpha\in R$. However, I've been told that an equivalent ...
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0answers
31 views

A consequence of “every norm is equivalent to the sup-norm” in a finite dimensional normed vector space

So, I am reading the following proposition in Neukirch's Algebraic Number Theory: Proposition. Let $K$ be a complete field with respect to the valuation $|\:\: |$ and let V be an $n$-dimensional ...
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37 views

Questions about the finite-dimensional normed space of polynomials of degree at most two.

Take $X:=P_2([0,1])$, the polynomials of degree at most $2$ over $[0,1]$ and consider the $2$-norm on this space. For any $x\in X$ we have that, $$\|x\|_2=\left(\sum_{i=1}^n|x_i|^2\right)^{1/2}$$ ...
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1answer
28 views

X and Y are normed linear spaces over the same field $F(=\mathbb{C}/\mathbb{R})$, both having the same finite dimension $n$.

X and Y are normed linear spaces over the same field $F(=\mathbb{C}/\mathbb{R})$, both having the same finite dimension $n$. I need to show that $X$ and $Y$ are topologically isomorphic ( A ...
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1answer
35 views

How does sketching norms show that they are equivalent?

I have the following statement in my notes: "You might want to check by drawing the sets of all $x\in\mathbb R^2$ such that $\|x\|_1=1$,$\|x\|_2=1$,$\|x\|_\infty=1$ that indeed these norms are ...
4
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0answers
31 views

Can We Always Realize the Value of the Quotient Norm. [duplicate]

Let $(V, \|\cdot\|)$ be a Banach space over $\mathbf R$ and $W$ be a closed subspace of $V$. We know that $V/W$ becomes a normed linear space under the quotient norm $\|\cdot\|_q$ defined as ...
2
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1answer
47 views

Understanding convergence in normed spaces and the language used when talking about norms.

We have the following definition about convergence in a normed space: "Let $(x_n)_{n=1}^\infty$ be a sequence in a normed space $(X,\|\cdot\|)$. We say that $x_n\to x$ in $X$ if, $$d(x_n,x)\equiv ...
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2answers
245 views

Examples of infinite dimensional normed vector spaces

In my notes on functional analysis it mentions that $C([0,1]),\ell^p$ and, $\ell^\infty$ are normed vector spaces, and gives some examples of norms that we can define on them. However, it then simply ...
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1answer
26 views

Need help with this proof, theory of finite change.

Theory: If $f : [a,b] \to X $ is differentiable on (a,b) and continuous on [a,b] in $X$, a normed vector space upon $ \langle , \rangle$ then: $$|f(b)-f(a)| \leq \sup_{a <c<b}{\|f'(c)\|}(b-a)$$ ...
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1answer
19 views

$| \langle a , i \rangle| \leq \| a\|$ if $\|i\|=1$ this space is a normed vector space upon $\langle , \rangle$ . Why does this apply?

I tried over Cauchy Schwartz to conclude, but could not. Anyone see why this is ? The term: normed vector space upon $\langle , \rangle$ i hear for the first time, Im assuming it means that: $$\|a \| ...
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30 views

Equivalence of Norms and Open Mapping Theorem

Let $V$ be a vector space with two norms $||\quad||_{1}$, $||\quad||_{2}$, making $V$ a complete normed vector space. Assume $\exists C$ (constant) such that: $||v||_{2} \leq C||v||_{1}, \forall v ...
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1answer
21 views

Operator norm and equivalent definitions

From the definition of the operator norm, we have: $||T||_{op}=\inf\{c\in \mathbb{R}^+:||Tv||\leq c||v||, v \in V\}$ If $T: V \rightarrow W$ is a linear map between two normed vector spaces. I have ...
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1answer
26 views

If a function $f: X \to Y $ $X, Y $ are some normed spaces, is continuous, then $\|f(x)\|< \infty , x \in X$ and why? [closed]

If a function $f: X \to Y $ $X, Y $ are some normed spaces, is continuous. Then is $\|f(x)\|< \infty , x \in X$ and why? I am thinking yes on this one (because otherwise, it would not be ...
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14 views

A metric space and polytopes

suppose we define a metric space $L_2(p)$ induced by a duality pair given by $\langle x,y\rangle =\sum_{j,k} p(j,k)[ x_1^j y_1^j+ x_2^k y_2^k]$ Where $x=(x_1,x_2)$ and $y=(y_1,y_2)$ and $p$ is a ...
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2answers
20 views

$f'(x;y)=0$ for every $x$ in an open convex set and for every vector $y$ ; then to show $f$ is constant on $S$

Let $f:\mathbb R^n \to \mathbb R$ be a map , $S$ be an open convex set in $\mathbb R^n$ such that for every $x \in S$ and $y \in \mathbb R^n$ , $f'(x;y)$ exists and equals $0$ ; then how to show that ...
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69 views

What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| ...
6
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1answer
67 views

Norm $\Vert \cdot \Vert$ on the symmetric group $S_n$

If we define a real valued function $\Vert \cdot \Vert$ on the $n^{th}$ order symmetric group $S_n$ satisfying following conditions $$\begin{align} & \|x\|=0\iff x=\omega\,\,\,(\text{identity ...
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1answer
28 views

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball (by closed ball I mean $B[a,r]:=\{x \in \mathbb R^n : d(x,a)\le r\}$ , ...
3
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1answer
26 views

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space?

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space? My textbook (Metric Spaces - Michael Searcoid) gives the following hints to answer the above ...
2
votes
1answer
37 views

Specific question on $l^p$ spaces and its dual in weak * topology

I am covering now Lp spaces in my summer real analysis course and this problem from Folland related to the dual of Lp stumped me hard, it is problem 19 chapter 6 reads as follows: We define $ ...
4
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2answers
36 views

Parallelogram law in $L_1$ space

Exercise 5.5 from Capinski's and Kopp's book "Measure, Integral and Probability" asks to show that it is impossible to define an inner product on the space $L^1([0,1])$. In order to get this result we ...
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0answers
25 views

Generate complete functions set in hilbert space

It is just a curiosity, but is there a general method (or a class of methods) that allows to derive orthonormal complete function set for a given hilbert space? (Except Gram Shmidt algorithm and ...
2
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0answers
14 views

How are $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ identified as Banach algebras?

When $A$ is a Banach algebra, one can make $M_n(A)$ into a Banach algebra by equipping it with various norms. One can also make the algebraic tensor product $M_n(\mathbb{C})\otimes A$ into a Banach ...
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1answer
36 views

Is $C[0,1]$ equipped with $\lVert \cdot \rVert_1$ a countable union of nowhere dense sets?

Let's consider the space of all continuous function $C[0,1]$ on the intervall $[0,1]$. But instead of using the usual supremum norm we use the $L^1$-Norm $: \lVert f \rVert_1=\int_0^1 \lvert f(x) ...