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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38 views

Is there only one way to define a norm from an inner product?

Given an inner product $\langle,\rangle$, we can define a norm by $||x|| = \langle x,x \rangle^{\frac{1}{2}}$. My question is, are there other ways to derive a norm from an inner product space and if ...
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1answer
54 views

Proof of inequality in a normed space

Let $E$ be a normed space and let a vector $u\in E$ and a vector $v\in E$ with $||v||=1$ that satisy $||u+v||\geq 2-\varepsilon $ for a $\varepsilon >0$ be given. Show that for all $\alpha ,\beta &...
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28 views

Will the problem right?

Let $X$ be a normed space and $Y_1$ , $Y_2$ complete spaces and exist two injective linear application $f_1:X \to Y_1$, $f_2:X\to Y_2$ such that $\text{closing } f_1(X)=Y_1$, $\text{closing } f_2(X)...
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1answer
46 views

Can someone explain why $\| \cdot \|_p, p = \frac{1}{2}$ isn't a norm?

$\| \cdot \|_p, p = \frac{1}{2}$ or "half norm" is not a norm What is a quick way to verify that it is indeed not a norm?
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49 views

Why in normed vector spaces we can define infinite series but in metric space we can not?

We usually define infinite series by partial sums and an inifinite series is said to converge if its partial sum converges. So, if $X$ is a normed vector spaces and $s_n=x_1+...+x_m$ is a partial sum ...
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164 views

What is a “pseudonorm”?

The following is an excerpt of a note in topological vector spaces. I have tried to search "semi-pseudonorm" on Google but I have got nothing so far. A search with "pseudonorm" returns what we ...
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15 views

Translation of a slice inside the unit ball

Let $B$ the unit closed ball of a $\mathbb R$-normed space $E$, $\ell$ a continuous linear form and $u\in E$ such that $\Vert u\Vert=\Vert \ell\Vert=\ell(u)=1$. Let $H=\lbrace x\in E\,; \ell(x)\ge\...
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1answer
33 views

Quickly sketching the power function $x^{2/3}+y^{2/3}=1$

What is the best way to quickly sketch $x^{2/3} + y^{2/3} = 1$ by hand, without using a graphing device? One can quickly imagine that $x^2 + y^2 = 1$ is a circle. But how does one quickly imagine ...
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44 views

Absolutely convergence property on normed spaces implying continuity of linear operator

Let $X,Y$ be normed spaces and $f:X\to Y$ a linear operator. Suppose $f$ is such that $\sum_{n=1}^\infty f(x_n)$ is convergent in $Y$ whenever $\sum_{n=1}^\infty \|x_n\| < \infty$. With this ...
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41 views

An equality in normed space

Let $X$ be a normed space. Suppose that for some $x, y \in X$ we have $\lVert x+y \rVert = \lVert x \rVert + \lVert y \rVert$. Prove that $\lVert \alpha x + \beta y \rVert = \alpha \lVert x \rVert + \...
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63 views

Why is the norm ball a square in $\,\mathbb R^2\,$ under $\,l^\infty\,$ norm?

Suppose $\,x = \left(x_1, x_2\right)$, then $\,l^2\,$ norm ball is $\,\left\lbrace x\;\big\vert\;\, \sqrt{\left\lvert x_1 \right\rvert^2 + \left\lvert x_2 \right\rvert^2} \leq 1\right\rbrace$ Easily ...
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0answers
55 views

Is the norm ball a set or the boundary of a set?

Recall normed ball in $R^2$ under different norms is typically intuited as follows But looking at someone of the definition of normed ball it seems that it describes a closed set rather than the ...
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1answer
17 views

$inf\{||x|| : x \in X , ||F(x)|| = \alpha\} = \frac{\alpha}{||F||}$

If $F \in B(X,Y), F\neq 0$ and $\alpha \geq 0$, then show that $$inf\{||x|| : x \in X , ||F(x)|| = \alpha\} = \frac{\alpha}{||F||}$$ where $B(X,Y)$ is the set of all bounded functions from $X \to Y$ ...
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49 views

Convolution inequality, Gaussian convolution

I'm reading a proof that says for $f\in L^p$ with $p\in[1,\infty)$ we have $\|f\ast p_t-f\|_p\to 0$ as $t\to 0$, where for $t>0$, $p_t(x)=\frac{1}{(2t\pi)^{\frac{d}{2}}} e^{-\frac{\|x\|^2}{2t}}$ is ...
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0answers
29 views

Norm of homomorphism embedding Banach algebra into matrix algebra

Let $A$ be a nonunital Banach algebra, and let $\tilde{A}$ be the unitization of $A$, i.e. $\tilde{A}=\{(a,\lambda):a\in A,\lambda\in\mathbb{C}\}$ with multiplication given by $(a,\lambda)(b,\mu)=(ab+\...
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88 views

Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
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1answer
87 views

Difficulty understanding the proof of equivalence of all norms over $\mathbb R^{n}$

To prove that all norms are equivalent on $\mathbb R^{n}$ , the book I am reading , first takes an arbitrary norm $$|\ \ | \ :\ \mathbb R^{n}\rightarrow\ \mathbb R$$ and then ...
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2answers
99 views

In the proof that $L^{1}$ norm and $L^{2}$ norm are equivalent.

To prove that $L^{1}$ norm , denoted by $||\ \ ||_{1}$ and $L^{2}$ norm , denoted by $||\ \ ||_{2}$ are equivalent we have to find constants $C_{1},\ \ C_{2}$ that satisfies ...
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35 views

Prove that $\| x \| \le \alpha {\| x \|_{sum}}\,,\,$ $\forall x\in R^n\;$ such that $\alpha=\max_{1\le i\le m}\{\|e_i\|\}\,,\;$ $\|\cdot\|=$ any norm.

Prove that $$\| x \| \le \alpha {\| x \|_{sum}}\quad,\quad \forall x\in \mathbb{R}^n$$ for any norm $\| {\, \cdot \,}\|\quad$ , $\quad{\left\| x \right\|_{sum}}$ is the norm of the sum $\quad$, $\...
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1answer
45 views

Strategy for establishing the triangle inequality of a seminorm

One proof that the $p$-norm $\| x\|_p = (|x_1|^p + \ldots + |x_n|^p)^\frac{1}{p}$ satisfies the triangle inequality exploits the fact that $ x \mapsto |x_1|^p + \ldots + |x_n|^p$ is a convex ...
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38 views

How to show continuity of $\cdot$ in a normed vector space?

Let $(V,+,\cdot)$ be a normed vector space with the underlying field $K$ . We have to show that $+,\cdot$ are continuous functions. Since $V$ becomes a metric space under this norm so I can use ...
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1answer
55 views

Equivalent norms on $C[0,1]$

For each $f\in C[0,1]$ set $$\|f\|_1 = \left(\int_0^1 |f(x)|^2 dx\right)^{1/2},\quad\quad \|f\|_2 = \left(\int_0^1 (1+x)|f(x)|^2 dx\right)^{1/2}$$ Then prove that $\|\cdot\|_1$ and $\|\cdot\|_2$ are ...
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34 views

Compute the norm of the operation $A$

Suppose that $\left ( a_{ij} \right )_{i,j=1}^{\infty}$ is a matrix satisfying the following condition $$\sum_{i,j=1}^{\infty} \left | a_{ij} \right |^q < \infty$$ where $q>1$. For $x=\left \{ \...
2
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1answer
56 views

How strong is the operator norm topology?

Let $(V,\tau_V), (W,\tau_W)$ be normable topological vector spaces. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ inducing $\tau_V, \tau_W$ respectively. Let $||\cdot||_{op}$ be the operator norm ...
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35 views

Norms on $L(V,W)$

Let $V,W$ be normable topological vector spaces over $\mathbb{F}$. Let $C(V,W)$ be the set of continuous linear transformations $T:V\rightarrow W$. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ ...
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252 views

Operator Norm of a Linear Transformation of a Matrix

The book I am using for the ODE course is Differential Equations and Dynamical Systems by Lawrence Perko. I am having a difficult time understanding what an operator norm of a linear transformation ...
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1answer
27 views

Finding an element in $l_1$ space with certain properties

I am facing a bit problem in the following: Given $x_1,...,x_m \in l^\infty$ and positive $\epsilon_1,...,\epsilon_m$, I need to find an element $a= (a_n)$ in $l_1$ space such that $\sum_{n=1}^ \infty ...
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0answers
44 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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1answer
73 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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110 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 ...
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2answers
43 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
216 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
65 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
79 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
76 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
48 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
32 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
43 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
46 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 ...
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2answers
33 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
38 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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47 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 $C^1([...
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2answers
75 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
32 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 $\{x_n\}$ ...
4
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2answers
90 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
45 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
165 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
40 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
votes
1answer
137 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 p\lt\...
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1answer
64 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 ...