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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1answer
66 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 ...
3
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2answers
73 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
12 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
31 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 ...
2
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1answer
58 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
9 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
73 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
29 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. $\pi R=\text{Id}_{V/W}$ ...
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1answer
49 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) ...
2
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1answer
48 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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1answer
31 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
51 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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1answer
42 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
140 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
50 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
32 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
97 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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2answers
100 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
88 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
30 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
64 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
75 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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2answers
91 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 $\phi(-2y,y,z)=-3y$, ...
2
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1answer
43 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\{ ...
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1answer
68 views

Check if a function is L2

I want to check if a function $f$ defined on $[0,T]$ is a $L_2$ function. What I know is $f$ is a $L_1$ function. (but $f$ could be not bounded) So I want to use an inequality like $$ ...
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1answer
65 views

Open mapping lemma - are these versions equivalent?

Here is a version the Open Mapping Lemma given in class : Let $X$ be a Banach space and $Y$ be a normed space. Let $T : X\rightarrow Y$ be a bounded linear map. Assume there exist $M \geq 0$ and ...
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2answers
76 views

Prove that $T^*$ is injective iff $ImT$ Is dense

Let X,Y be two normed spaces, and $T:X\rightarrow Y$ a bounded linear operator. prove that the adjoint operator $T^*$ ($T^*f(x)=f(Tx)$ is injective iff $ImT$ is dense any help would be great guys. I ...
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0answers
39 views

Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm?

Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm? Suppose that $\lambda _n \to \lambda $, $\mu _n \to \mu ...
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6answers
628 views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
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0answers
113 views

Commutativity and norms of specific operators (Problem 2.7.10 in Kreyszig's functional analysis book)

This is Problem 2.7.10 from Erwin Kreyszig's Introductory Functional Analysis with Applications. Let $C[0,1]$ denote the normed space of all (real- or complex-valued) functions defined and ...
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1answer
32 views

Show whether $N(g)=\|g\|_\infty + \|g'\|_\infty$ and $\|\cdot\|_\infty$ are equivalent on $C[0,1]$

Let $L$ be the linear subspace of $C[0,1]$ is the space of continously differentiable functions. I know I've got to show whether there exists an $a,b>0$ such that: $$a\|x\|_\infty \leq ...
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2answers
119 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Problem 8, Section 2.7

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...
5
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1answer
289 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.7, Problem 9

Here is Problem 9 in the Problem Set following Section 2.7 in the book Introductory Functional Analysis With Applications by Erwine Kryszeg: Let $C[0,1]$ denote the set of all (real- or ...
3
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2answers
45 views

Intersection of compact convexes

Let $C_1,C_2,C_3,C_4$ be compact convexes of $\mathbb{R}^2$ such that $C_1\cap C_2\cap C_3\neq\emptyset,C_1\cap C_2\cap C_4\neq\emptyset,C_1\cap C_3\cap C_4\neq\emptyset,C_2\cap C_3\cap ...
3
votes
2answers
96 views

Metrizability of the unit ball $B_{X^*}$.

I am trying to prove the assertion: If $X$ is a separable normed space, then the unit ball in $X^*$ with the weak* topology, $(B_{X^*},\sigma(X^*,X))$, is metrizable. Firstly, I took ...
3
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1answer
87 views

Closed subspace. A Hahn–Banach theorem consequence

I am trying to prove: If M is a subspace of a normed space $X$, that $\overline{M}=\bigcap\{\ker(\phi):\phi|_{M} = 0 \}$ It is really easy to see that $\overline{M} \subset ...
2
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0answers
52 views

Norm for a set of vectors

Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$ I ...
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1answer
20 views

Product of $L_2$ norm of vectors

Is the $\sum \Vert b_k\Vert_2^2 \le\ge= \sum \Vert b_k\Vert_2^2 \Vert a_k\Vert_2^2$ ? where $b_k$ is a column vector and $a_k$ is a highly sparse row vector.
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1answer
145 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.7, Problem 6

Suppose that $X$ and $Y$ are two normed spaces over the same field ($\mathbb{R}$ or $\mathbb{C}$). Show that the range of a bounded linear operator $T \colon X \to Y$ need not be closed in $Y$. ...
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1answer
57 views

If the dual unit ball of a normed space $X$ is metrizable in the weak-$*$ topology then $X$ is separable

Let $X$ be a normed space and $(B_{X^*},w^*)$ be the unit ball of the dual space $X^*$ endowed with the weak-$*$ topology. Here is a proof a the fact that if $(B_{X^*},w^*)$ is metrizable then $X$ is ...
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1answer
31 views

Unit ball for a special norm

What does the unit sphere for the norm on $\mathbb{R}^2$, $\displaystyle N(x,y)\rightarrow\sup_{t\in\mathbb{R}}\frac{|x+ty|}{t^2+t+1}$, look like ? My approach was to consider $y=ax$ so as to get ...
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0answers
14 views

Problem of Closed linear transformation in Normed spaces [duplicate]

Let $X$ a normed space and let $A$ and $B$ be linear transformations such that $$X\subset D_A\rightarrow^{A} X \ \ \text{and} \ \ X\subset D_B\rightarrow^{B} X.$$ If $A$ and $B$ are closed, does it ...
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3answers
1k views

Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my ...
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1answer
42 views

How to prove that operators are isometry on $\ell^p$/$c$?

Lately I've been studying Banach spaces and isometries, and encountered many explicit isometrys involving $c$, $c_0$, $c^*$, $c_o^*$, $\ell^1$, $\ell^\infty$, etc. (Here $c\subset\ell^\infty$ is the ...
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2answers
40 views

Is a countable union of complete subspaces complete?

I would like to ask the following, which I wanted to use a part of my proof but couldn't determine if it's right: Assume $X$ is a normed space, and $(X_n)_{n\in \mathbb N}$ complete subspaces. Must ...
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1answer
14 views

Boundedness of multivariable polynomials

How can we prove multivariable polynomials are bounded on a closed set? the boundedness theorem is for single variable functions. Does an extension theorem exist? Thank you.
1
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2answers
64 views

Closed AND open subspaces of a normed vector space

Let $E$ be a finite dimension normed vector space. How can I show that $E$'s only both closed and open (norm-wise) subsets are $\emptyset$ and $E$ ?
0
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1answer
40 views

The exponential of the identity operator in a Banach space

Let $X$ be a Banach space and $I \in L(X)$ be the identity operator. Determine the action of the operator $e^I$ on $X$. Pretty stuck here, not sure exactly what it means by determine the action. ...
0
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0answers
26 views

Norm of linear functional in $W^{1,2}$

I want to find the norm of the following linear functional: $\phi:W^{1,2}_{[0,1]} \rightarrow\mathbb{R}$ $\phi(f)=\int^1_0t\cdot f(t)+ f^{'}(t) dt $ The norm is: ...
0
votes
0answers
67 views

Riemann integrals of abstract functions into Banach spaces

If we define the (Riemann) integral of an abstract function, i.e. a function $f:[a,b]\to Y$ where $Y$ is a Banach space, as$$\int_a^b F(t)dt:=\lim_{\max(t_{k+1}-t_k)\to ...