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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21 views

>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
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19 views

Show that the application $ Id:( C^1(0,1), \|\cdot\|_{1})\to ( C^1(0,1), \|\cdot\|_{\infty})$ is not continous

I want to prove that the application $$ Id:( C^1(0,1), \|\cdot\|_{1})\to ( C^1(0,1), \|\cdot\|_{\infty}) $$ is not continous. If I prove that this application is not bounded I have finished. So I ...
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0answers
31 views

n- dimensional normed linear space isomorphic to n-dimensional Euclidean space

If $(X,\|\cdot\|)$ is an n- dimensional normed linear space over R. Is it isomorphic to n-dimensional Euclidean space $R^n$. I know it is topologically isomorphic but what about isometry? I think if ...
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5 views

Covering Number of Lipschitz Function Space

On page 12 of the slide deck here, the author gives an example where a lower bound (and upper bound, but I am particularly interested in the lower bound) on the $\epsilon$-covering number of a ...
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1answer
32 views

Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
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1answer
18 views

Does a normed space with norm2 defines an inner product?

I know that generally, an inner product defines a norm on an inner product space, But, generally speaking, If I have a normed space (on purpose I do not say which) with the norm 2 does it mean that I ...
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1answer
35 views

Bounded functional which composed with an unbounded operator becomes unbounded

While working on a problem, I came up with a certain lemma, however I'm not sure whether it's true and I'd be grateful for some insight. Let $ X $ and $ Y $ be normed spaces over reals, where $ X $ ...
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19 views

weak completness and infinite dimensional spaces

I try to prove: An infinite-dimensional normed linear space is never weakly complete. First part of this exercise is to prove that: A normed linear space X is finite dimensional if and only if every ...
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1answer
33 views

Isometry under condition $\|x-y\|=n\iff\|f(x)-f(y)\|=n.$ for $n\in\mathbf{N}$

Let $X, Y$ be normed spaces and $f:X\to Y$ be mapping and $n\in\mathbf{N}$ If$$\|x-y\|=n\iff\|f(x)-f(y)\|=n.$$ Under what conditions this map will be an isometry? Thanks
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Prove the triangle inequality is valid for the norm $\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$

I.e, prove $\lVert f+g \rVert\ \le \lVert f \rVert + \lVert g \rVert$ for all $f,g$ in $C^\infty [0,1]$, $$\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$$ I think we're supposed to use ...
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45 views

Prove that L 2 PC[−1, 1] is not a complete normed space

I'm trying to prove that the normed space of all piecewise continuously functions with the norm $$\int^1_{-1}|f(x)|^2dx$$ is not a complete normed space. $L_2PC[-1,1]$ for that, im trying to find a ...
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1answer
45 views

Why is $\langle x-P(x),m\rangle=0$?

Let $H$ be a Hilbert space, and let $M\le H$ be a subspace of it. Let $P:H\rightarrow M$ be the orthogonal projection $H$ onto $M$. We'll take $x\in H$, and $m \in M$. By the definition I know ...
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16 views

Why is a linear transformation of a cauchy sequence in a normed space also cauchy?

Suppose we have a cauchy sequence $\{a_n\}$ in a normed vector space $V$. Given a linear transformation $T:V \rightarrow V$, is the sequence $\{T(a_n)\}$ also cauchy? Or is it true only for finite ...
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13 views

Is the projection of a cauchy sequence in a normed finite dimensional vector space along some subspace also cauchy?

Let $V$ be a normed finite dimensional vector space. Let $S$ and $S'$ be two subspace such that $S \cap S'={0}$ and $V$ is the direct sum of $S$ and $S'$. We define the projection of a vector $x$ ...
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2answers
26 views

Is the norm of a projection of a vector along a subspace less than or equal to the norm of the vector iteself?

My question is, given a vector $x$ in a normed space, and any two subspaces with an intersection $0$ and whose direct sum is the whole vector space, is the norm of projection along one of the ...
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1answer
24 views

Is there a point that lies on the boundary of the unit ball in $\lVert\cdot\rVert_1$, and close to the zero-sequence in $\lVert\cdot\rVert_2$?

I am an engineer who is brushing up some functional analysis. I am curious about the following problem I posed to myself: Consider the sequence space of real-valued sequences that will eventually ...
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1answer
18 views

Showing a function to be a norm

I want to prove or disprove that $\parallel (x,y)\parallel=\sqrt{\frac{x^2}{9}+\frac{y^2}{4}}$ is a norm on $\mathbb{R^2}$. Since $\{(x,y):\parallel(x,y)\parallel\leq1\}$ is a convex set, ...
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2answers
141 views

Can we have something like an orthonormal basis for a finite dimensional normed space?

So I proved a certain theorem about finite dimensional inner-product spaces, but after completing the proof, I realized the only point where I used the idea of orthogonality was the construction of an ...
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2answers
36 views

Equivalence of norms on vector spaces

Let us call two norms $|x|_1$ and $|x|_2$on a finite-dimensional vector space equivalent if they set the same topology on that space. I need to show that this definition is equivalent to the existence ...
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19 views

Distance to a closed ball in a normed space.

Let $(E, \|\cdot\|)$ be a normed vector space, and consider $B = B[{\bf a},r]$ the closed ball. Let ${\bf b}\in E$. Then $\newcommand{\d}{{\rm d}} \d({\bf b},B) = 0$ if and only if ${\bf b} \in B$. ...
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31 views

Geometrical meaning of a face

Let $(X,P)$ be a locally convex space, $K$ a compact, convex subset of $X$. A face $F$ of K is a nonempty, compact, convex subset of $K$ s.t. $$\forall y,z\in K \,\forall t\in(0,1) \left[ (1-t)y + tz ...
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1answer
26 views

closure of a convex set in a normed linear space is convex ?

Is it true that if $A$ is a convex set in a normed linear space $V$ , then the closure of $A$ is also convex ? (I know that the interior is convex )
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2answers
43 views

Is $T:C[0,1]\to C[0,1]: Tf(x)=\frac12 x f(x^2)+1$ a contraction?

I am a non-mathematician (a physicist) who wants to find a continuous function on $[0,1]$ that satisfies the relation $f(x)=\frac12 x f(x^2)+1$ with $0\le x\le 1$. I need it for one of my models. ...
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3answers
112 views

Compactness of subspaces of finite-dimensional vector spaces

For subspaces of $\mathbb R^n$ we know that they are compact if and only if they are closed and bounded. Is the same true for all normed finite-dimensional vector spaces perhaps not over $\mathbb R$ ...
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2answers
53 views

A condition equivalent to orthogonality

Prove that in any inner product space: $x$ and $y$ are orthogonal if and only if $||x+\alpha y||\ge ||x||$ for every scalar $\alpha$.
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1answer
59 views

Question about a proof in Rudin's book - annihilators

I'm reading the proof of the following theorem in Rudin's "Functional Analysis": Let $M$ be a closed subspace of a Banach space $X$. The Hahn-Banach theorem extends each $m^* \in M^*$ to a ...
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1answer
49 views

How to use Triangle inequality to find the projection onto unit ball?

The projection onto the unit ball $$C:=\mathbb{B}(0,1)=\{x:||x||\leq1\}$$ is given by $$P_{C}(x)=\frac{x}{max\{||x||,1\}}, \quad\forall x\in X$$ where $X$ is Hilbert space. Now I can understand this ...
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51 views

How to prove a Banach normed vector space is NOT a Hilbert space?

We know that the Banach space $\big(\Bbb R^n,\|\cdot\|_2\big)$ is a Hilbert space with inner product $\langle x,y\rangle := \sum_{k=1}^n x_ky_k$. However, how to prove that $\big(\Bbb ...
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1answer
11 views

Is linear convergence norm invariant?

Let $\|\cdot\|_a,\|\cdot\|_b$ be two norms on $\Bbb R^n$ and $(x^k)_{k\in\Bbb N}\subset \Bbb R^n$ a sequence such that there exists $0<\alpha <1$ with $ \|x^{k+1}\|_a \leq \alpha \|x^k\|_a$ for ...
2
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0answers
13 views

Explicit Hahn-Banach extension formula in finite dimensional $l^p$ /Smoothness of the Hahn-Banach mapping

Consider the finite dimensional vector space $V=(\mathbb{R}^N,\|\cdot\|_{p})$, equipped with the usual $l^p$ norm, $1<p<\infty$. Consider a linear subspace $U\subset V$ (not necessarily a ...
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1answer
52 views

Which is finer(larger) between the sequence spaces $l_{p}$ & $l_{p+1}$

Prove that, $l_{3}\subset l_{7}$ & $L_{9}[0,1]\subset L_{6}[0,1]$, where $l_{p}$ & $L_{p}[0,1]$ are of usual notation. Are the converses hold for both cases? Can these two results ...
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1answer
7 views

balls have empty boundary with regard to the $p$-adic norm

Let $p$ be prime, $a\in\mathbb{Q}$ and $r\geq0$. How can I show that the closed ball $D(a,r)$ in $(\mathbb{Q},|\cdot|_p)$ must have an empty boundary (with regard to the topology induced by the ...
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1answer
71 views

about djoint operator

I'm trying to prove that if $X,Y$ are normed spaces and $T:X\to Y$ is linear operator (assume bounded, if needed), then $T^*$ linear. What i've been trying so far is taking $Tx_1,...,Tx_n$ a basis to ...
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1answer
23 views

Do projections (onto quotients by a closed subspace) have continuous right-inverses?

Let $V$ be a normed vector space and $W\subset V$ be a closed subspace. Does the projection $\pi\colon V\to V/W$ have a linear continuous right-inverse $R\colon V/W\to V$, i.e. $\pi R=\text{Id}_{V/W}$ ...
2
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1answer
41 views

if a point is a root of all linear functionals on a normed space, then it's zero

I've been trying to do some work as literal and detailed as possible in order to see I know my analysis for every detail. I tried to explain my self why if there is $x_0\in X$ ($X$ is a normed space) ...
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1answer
39 views

Norm of rotation matrix as element in $M_2(A)$

Let $A$ be a complex unital Banach algebra and let $R_t=\begin{pmatrix} \cos\frac{\pi t}{2} & -\sin\frac{\pi t}{2} \\ \sin\frac{\pi t}{2} & \cos\frac{\pi t}{2} \end{pmatrix}$. If I consider ...
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0answers
9 views

Proving linearity implies (or can imply under opportune conditions) lower semicontinuity

A function $f:X\to\mathbb{R}$, with $X$ being a topological space, is termed as lower semicontinuous (lsc) at $x_0\in X$ if: $$\forall\epsilon>0\,\,\exists V\text{ an open neighborhood of }x_0:x\in ...
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1answer
42 views

Why do we consider only real or complex Banach spaces?

In wikipedia, normed vector space is defined as a vector space over a subfield of $\mathbb{C}$ equipped with a norm. However, Banach space is defined as a complete normed space over $\mathbb{R}$ or ...
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22 views

Extreme points of complex sphere of dimension n in 1-norm.

I came up with the following question while learning about different norms in $\mathbb{C}^n$. For $z=(z_1, \ldots, z_n)^T \in \mathbb{C}^n$ we consider the 1-norm: $\|z\|_1= \sum_{k=1}^n|z_k|$. Let ...
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4answers
115 views

Isometry of a complete normed space is also complete.

Let $X$ be a complete normed space and assume the normed space $Y$ is isometric to $X$. Show that $Y$ is complete. I tried: Since X is complete $||x_n-x_m||<\epsilon, \forall n,m>N$ and since ...
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1answer
45 views

Prove $ \lim_{n\to+\infty}\int^{-\alpha}_{-1}|f_n(t)+1|dt=0$

Suppose $E$ a vector space of continuous function from $[-1,1]$ to $\mathbb{C}$, we define the norm: $$||f||_1= \displaystyle\int^1_{-1}|f(t)|dt$$ and we define a sequence such as: $$ f_n(t)= ...
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1answer
31 views

what is an example of a normed space such that $||\sum x_i||=\sum ||x_i||$ does not imply they have the same direction?

Let $V$ be a normed space over $\mathbb{K}$ and $x_1,...,x_n\in V\setminus\{0\}$ such that $||\sum_{i=1}^n x_i||=\sum_{i=1}^n ||x_i||$. If $||\cdot||$ satisfies parallel law, then this implies that ...
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1answer
40 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 4: How to show boundedness?

Let $f_1$, $f_2$ be the functionals defined on the normed space $C[a,b]$ of all continuous functions defined on the closed interval $[a,b]$ with the maximum norm be defined as follows: $$f_1(x) ...
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53 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 3: What is the norm of this functional?

What is the norm of the linear functional $f$ defined on the normed space $C[a, b]$ of all functions defined and continuous on the closed interval $[a,b]$ with the norm defined as $$\Vert x \Vert ...
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1answer
35 views

Erwin Kreyszig Section 2.8, Problem 2: What is the norm of these two functionals?

Let $a$, $b$ be two real numbers such that $a<b$, and let $C[a,b]$ denote the normed space of all (real- or complex-valued) functions defined and continuous on the closed interval $[a,b]$ with the ...
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1answer
25 views

Functional Analysis proof help involving decreasing sequence of sets in a finite dimensional normed space

Hi I am struggling with a proof; Prove that in a finite dimensional normed space, Z, any decreasing sequence of closed and bounded sets $\{C_i\}$ ($C_{i+1}\subseteq C_i\;\forall i$) cannot satisfy ...
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0answers
36 views

Gelfand width of $l_1$ unit ball in $\infty$ metric

The $l_1$ balls in $l^N_p$ are well studied, but I cannot find anything about estimates of $d^m(B^N_1,l^N_\infty)$ except of $d^m(B^N_1,l^N_\infty)\geq C\min\{1,\frac{\ln(eN/m)}{m}\}$ which is not ...
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1answer
58 views

In finite dimensional normed space, every convex set contains a basis

I've been reading Lemma 5.60 here: http://epge.fgv.br/we/MD/TeoriaEconomicaAvancadaI/2009?action=AttachFile&do=get&target=Aliprantis-Infinite-Dimensional-Analysis.pdf (p.g 200, =217 on the ...
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89 views

How can I calculate the norm of this linear functional on $\mathbb R^3$

I've been trying to calculate the norm of $\phi\colon \mathbb R^3\rightarrow \mathbb R$ defined by $\phi(x,y,z)=1.2x-0.6y$. I really don't know how to this. I did note that $\phi(-2y,y,z)=-3y$, ...
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1answer
34 views

Max norm of product normes spaces

Edit: I need some help with this. Let $$(V_1, \|·\|_1)$$ and $$(V_2, \|·\|_2)$$ be normed spaces, and the product space $$V = V_1\times V_2$$ be endowed with the norm $$\|(x_1, x_2)\| = \max\{ ...