Tagged Questions

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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0
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1answer
87 views

Isometry from Banach Space to a Normed linear space maps

Let $X$ be a Banach Space and $Y$ be a normed linear space. Show that if $T$ is an isometry then $T(X)$ is closed in $Y$. Let me have some idea to solve this. Thank you for your help.
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1answer
175 views

Proving that if $f$ is not continuous functional then $\ker f$ is dense

In the context of a first course in functional analysis I have seen the following exercise: Let $X$ be a normed space and $0\neq f$ a functional. Prove that if $f$ is not continuous then $\ker ...
1
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1answer
82 views

Finiteness of the dimension of a normed space and compactness

I am studying functional analysis, and in the setting of normed spaces I have seen the theorem that states that the unit ball is compact iff the space is finite dimensional. I also saw an exercise: ...
2
votes
1answer
110 views

Proving that $X/M$ is a banach space when $X$ is

I am trying to do an exercise in an introduction to functional analysis course: 1) Let $X$ be a normed space and $\{x_{n}\}_{n=1}^{\infty}\subseteq X$. Prove that $X$ is a banach space iff ...
5
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2answers
106 views

Linear operators on $C^\infty[a,b]$

I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? If we denote this space as ...
1
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1answer
110 views

Lipschitz condition in infinite dimensional vector spaces

If we have that $T:V \times W \rightarrow Y$ multilinear and $V,W$ are infinite-dimensional normed vector spaces.(the finite-dimensional proof is easy, since you can use compactness of the boundary ...
4
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2answers
464 views

Is the boundary of the unit sphere in every normed vector space compact?

I wanted to ask whether the boundary of the unit sphere in every normed vector space is compact? I know that this is true for simple examples, but how is it in general?
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1answer
29 views

Clarification about the space $C^0 ([-T,T],B)$

Let $B$ be a Banach space (in particular, $B$ is a function space equipped with the supremum norm). The space $C^0 ([-T,T],B)$ is the set of continuous functions on $[-T,T]$, valued in $B$. ...
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1answer
114 views

a normed vector space is normed closed iff it is weakly closed.

The claim is A subspace of a normed vector space is normed closed iff it is weakly closed. I can show one direction. Strong convergence implies weak convergence, so it is weakly closed. But I have ...
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0answers
39 views

bounded sets $ B\in X/M$ where $X$ is a normed vector space.

Let $X$ be a normed space, $ M\le X$ a linear subspace. Let $ X/M$ with the quotient norm. Prove that $ B \subset X/M$ is bounded iff there exist a bounded set $A\subset X$ such that $ B\subset [A]$. ...
2
votes
1answer
70 views

What is the standard (?) operator norm usually used in functional analysis?

I am studying introduction to functional analysis, in my lecture notes I have seen that a norm on functions is used in some proofs. For example I have seen the following: We note that for every ...
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1answer
287 views

proving that the quotient linear map of a continuous linear map is also continuous (normed spaces)

Let $X,Y$ be a normed vector spaces over $\mathbb k $, $T:X\to Y$ a $\mathbb k$-linear continuous map ($\mathbb k$ could be $\mathbb R$ or $\mathbb C$). Let's consider $ \hat T: X/Ker T \to Y$ the ...
0
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1answer
81 views

Difference: normed space and normed linear space.

I'm currently reading up on Banach spaces. My book "Introduction to Banach spaces and their Geometry" by Beauzamy mentions "normed spaces" in some places, and "normed linear spaces" in other. I really ...
3
votes
1answer
68 views

Space of Differential Operators

Is is possible to talk about a "space of differential operators"? If one defined such a space would it be possible to talk about limits? I don't really have much background so I'm really just looking ...
3
votes
1answer
127 views

Equivalent conditions for weak and weak-$*$ convergence

Say $E$ is a normed space over the field $\mathbb K$ ($\mathbb R$ or $\mathbb C$) and $E^{*}$ its dual space. The notations for weak and weak - $*$ convergence are $x_{n} \xrightarrow{w} x$ and ...
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0answers
262 views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
0
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1answer
98 views

normed vector space real analysis

Prove that $\lVert x\rVert = \left(\sum_{k\in\mathbb{N}} \lvert x_k\rvert^p\right)^{1/p}$ is not norm for $\ell^p = \{x = (x_k)_{k\in \mathbb{N}} : \sum_{k\in\mathbb{N}} \lvert x_k\rvert^p < ...
1
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1answer
63 views

A question about the quotient topology in normed linear spaces.

Say $M$ is a closed linear subspace of normed linear space $N$. The coset of the form $x+M, x\in N$ in the quotient space $N/M$ is defined by $$\|x+M\|=\inf\{\|x+m\|:m\in M\}$$ Let us consider the ...
1
vote
1answer
202 views

Exercise in Tao Analysis Book

I'm currently studying in the book of Analysis of Terry Tao, amazing book by the way. In one exercise I'm not pretty sure about how can do it (I know that will be almost trivial but I'm stuck in it). ...
3
votes
1answer
77 views

Norm equality in the dual space

Suppose $X$ is normed complex space and $h:X\to \mathbb{R}$ is bounded linear functional (real). Prove that $f:X\to \mathbb{C}$ defined by $f(x)=h(x)-ih(ix)$ belongs to the dual space of $X$ and ...
2
votes
1answer
206 views

Prove that two norms are equivalents

Two norms $\|\bullet\|_1$ and $\|\bullet\|_2$ are equivalents iff $\;\exists\;c_1,c_2>0$ such that $c_1\|x\|_1\le \|x\|_2\le c_2\|x\|_1$ We're working in $\mathcal C^1[0,1]$, and I have ...
14
votes
4answers
428 views

If two norms are equivalent on a dense subspace of a normed space, are they equivalent?

Given a vector space $V$ equipped with two norms $|\cdot|$ and $||\cdot||$ which are equivalent on a subspace $W$ which is $||\cdot||$-dense in $V$, are the two norms necessarily equivalent? The ...
1
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1answer
92 views

Triangle inequality question on norm space

I'm trying to decide if $||v||=x^2+y^2$ defines a norm on $\Re^2$. It's been a long time since I prove normed spaces so please excuse me by being a rookie. 1) I'm having trouble specifically trying ...
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0answers
41 views

Norm of a mapping

$$C[0,1]=\{f:[0,1]\rightarrow R | \text{$f$ is continuous function}\}$$ $$\|f\|=\max \{|f(x)|:x\in [0,1]\}$$ $$A:(C[0,1],\|\|)\rightarrow(C[0,1],\|\|)$$ $$A(f)(x)=(x^4-x^2)f(x)$$ I have to ...
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0answers
63 views

The dual space of a nonempty normed linear space is non empty

Is the statement true? The dual space of a nonempty normed linear space is non empty? I am not able to prove or disprove, could anyone give me just hints? I know that it will be a norm linear space ...
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2answers
116 views

A problem on the bounds of Lp-norms

Let $L>0$ and $\Omega$ be the set of all integrable functions from $[0,L]$ to $[0,+\infty]$. Also, Let $f\in \Omega$ such that $\left \| f \right \|_{1}=1$. Find the tightest possible bounds for: ...
0
votes
1answer
38 views

Is $D$ well-defined?

In my text there's a problem which reads as: Consider $C[0, 1]$ with the norm $\|.\|_\infty$. Let $Y$ be the vector subspace of all differentiable functions on $[0, 1].$ Consider the linear map ...
3
votes
1answer
60 views

Let $T:X\to Y$ be continuous at $0.$ Then $\exists~k>0$ such that $\|Tx\|<k\|x\|.$

Let $T:X\to Y,~(X,Y$ being Normed Linear Spaces$)$, be a linear transformation continuous at $0.$ Then $\exists~k>0$ such that $\|Tx\|<k\|x\|.$ My attempt: $T$ is continuous at $0\implies$ for ...
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0answers
136 views

Proof that normed space is Banach space

I have to prove that $(l^{\infty},\|\cdot\|_{\infty})$ is Banach space and I have some difficulties. This is what I've done. $l^\infty=\{x=\langle x_k\rangle, k\in N|\exists M>0 \ such\ ...
2
votes
1answer
54 views

$(P[0,1],\|\|_{\infty})$ be the norm linear space

Let $(P[0,1],\|\|_{\infty})$ be the norm linear space and $T$ be the differentiation operator on it. Then $1.$ $T$ is onto right? but NOT injective as $\ker T=\{\text{ all constants }\}$ $2.P[0,1]$ ...
0
votes
0answers
44 views

$E_1+E_2$ is open if both open?

if $X$ be a norm linear space and $E_1,E_2\subseteq X$ then $E_1+E_2=\{x+y:x\in E_1,y\in E_2\}$ is open if both open? is open if one is open and another is closed? closed if both are closed? I just ...
10
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1answer
220 views

Proof that multiplying by the scalar 1 does not change the vector in a normed vector space.

I'm beginning a self-study of functional analysis, and I seem to have come to a halt trying to solve the first problem in the first problem set, and was wondering if someone could give me a pointer. ...
0
votes
2answers
44 views

how to show $\|T\|\le 1$

Given that $M$ is a closed linear subspace of $N$ and if $T$ is a natural mapping of $N\to N/M:x\to x+M$, I have shown that $T$ is continuous , but I am not able to show $\|T\|\le 1$ Thank you for ...
2
votes
1answer
75 views

Example of infinite dimensional B* space where weak convergence does imply strong convergence

So I know that weak convergence does imply strong convergence if the dimension of the space is finite, and that in general it does not in infinite dimension. But I was wondering if there were any ...
1
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1answer
139 views

A distance-minimizing continuous projection onto a finite-dimensional subspace?

Let $E$ be a Banach space, which need not be a Hilbert space, and let $F$ be a finite-dimensional subspace of $E$. Suppose that for all $x \in E$, there exists a $y \in F$ realizing the minimal ...
3
votes
2answers
22 views

Density and the size of coefficients

Let $E$ be a Banach space, and $F$ a dense subspace spanned by a countable base $y_i$ of unit norm. Let $x \in E$ and $x_n = \sum_{i_n=1}^{N_n} a_{i_n} y_{i_n}$ be a sequence of elements of $E$ ...
3
votes
1answer
82 views

Do I have a Banach space given the following norm?

This is very very similar to my other question asked three months ago. That time there was no Banach space because an integral in the norm definition allowed a counter-example. Once again I have a ...
0
votes
1answer
47 views

$\mathbb C$ over the field $\mathbb R$ or $\mathbb C$ over the field $\mathbb C$

When we talk about the topology of the complex plane what type of $\mathbb C$ as a normed linear space we get concerned about viz. $\mathbb C$ over the field $\mathbb R$ or $\mathbb C$ over the ...
2
votes
2answers
89 views

Generalizing norms for modules

An normed space is defined as a vector space V plus a norm operation over $V$. Is it meaningful to generalize this notion to modules, where one is dealing with rings instead of fields? What I'm ...
1
vote
1answer
166 views

How to show Zygmund space is Hölder space?

The motivation of this question is to show that Zygmund space is Hölder space, in certain cases. For simplicity, take $s\in (0,1)$, I want to show $$\|f\| = \|f\|_\infty + \sup_{x,y\in ...
0
votes
1answer
118 views

Showing a norm preserving isomorphism of vector spaces

Lets define $l^1$ as the complex vector space of all absolutely summable sequences of complex numbers and and $x_0$ consists of all the sequences in $l^\infty$ (all bounded sequences) that eventually ...
1
vote
2answers
59 views

how to show that $A_kB_k\to AB?$

Let in the space $M(n,\mathbb R)=$ set of all $n\times n$ real matrices endowned with $\| \cdot \|_2,~A_k\to A,~B_k\to B.$ Then how to show that $A_kB_k\to AB?$
2
votes
0answers
68 views

Distance from a point to a plane in normed spaces

How can we calculate the distance from a point to a plane in normed spaces ? where we are not inner product, for example: Calculate the distance in $(C[0,1],||\cdot||)$ endowed with the supremum norm ...
2
votes
1answer
289 views

How to proof homeomorphism between open ball and normic space

How can I prove that an open ball $B$ in a normed vector space $X$ is homeomorphic to $X$?
5
votes
1answer
564 views

Prove the boundedness of a bilinear continuous mapping.

Let $X,Y,Z$ are Banach spaces and $$B:X\times Y\to Z$$ is bilinear and continuous. Prove that there exists $M<\infty$ such that $$\lVert B(x,y)\rVert \leq M\lVert x\rVert\lVert y\rVert.$$ Is ...
1
vote
0answers
61 views

Definition of a projection on a normed space? Banach space?

Given a vector space $V$, a projection $V\to V$ is an idempotent linear map. For a normed space do we require anything else of the definition like continuity? Is the image required to be closed in ...
1
vote
1answer
39 views

Distance of a function from a subspace

Let $f \in L^2([-a,a])$. Trying to find $\mathrm{dist}(f,S)$ in $L^2([-a,a])$ (where S is the subspace of real polynomials of max degree $2$, like $a+bx+cx^2$) and knowing that $\langle f,a\rangle=0$ ...
0
votes
1answer
101 views

Determinant of Schur Complement

If I have an $n \times n$ real-valued non-symmetric matrix $\mathbf{M}$, which has determinant $|\mathbf{M}| > 0$, what can I say about the determinant of the matrix $\mathbf{Q}^T \mathbf{M}^{-1} ...
1
vote
2answers
67 views

Is't a correct observation that No norm on $B[0,1]$ can be found to make $C[0,1]$ open in it?

There's a problem in my text which reads as: Show that $C[0,1]$ is not an open subset of $(B[0,1],\|.\|_\infty).$ I've already shown in a previous example that for any open subspace $Y$ of a ...
3
votes
1answer
817 views

Weakly compact implies bounded in norm [duplicate]

The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous. If a subset $C$ of $X$ is compact for the weak topology, then ...