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.

learn more… | top users | synonyms

2
votes
1answer
123 views

uniform convergence on compact subsets of the linear,continuous and uniformly bounded operators.

Let $X,Y$ be normed spaces. Let $T_j : X\to Y$ be a sequence of linear and continuous functions, such that $\lVert T_j\rVert\lt K$ $\forall j$. If $T_j$ converges pointwise to $T$, prove that $T$ is ...
1
vote
0answers
51 views

Show that $S_x$ contains an interval

$X$ is Normed vector space, and $S_x={\{x\in X\space|\space\space\space||x||=1\}}$. Prove that if exist $x,y\in X$ which are linearly independent such that $||x+y||=||x||+||y||$, hence exist $u,v\in ...
0
votes
1answer
27 views

Norms on $\mathcal{M}_{n \times n}(\mathbb{R})$

On the space of all matrices $n \times n$ with real coefficients $\mathcal{M}_{n \times n}(\mathbb{R})$ we define two norms: $||A||_1 := sup_{x\neq 0; x \in \mathbb{R}^n} \frac{||Ax||}{||x||} \\ ...
4
votes
2answers
153 views

Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.

Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed. I'm having trouble in ...
1
vote
1answer
61 views

Series constructed from a cauchy sequence

Given a cauchy-sequence $\{x_i\}_{i\in \mathbb{N}}$ in a normed space $X.$ I need to construct a series that converges in $\mathbb{R}$ with $\{y_i\}_{i\in \mathbb{N}}$ a sequence in $X$: ...
2
votes
1answer
348 views

partial derivatives continuous $\implies$ differentiability in Euclidean space

I am given this theorem: If $f \in C^1(A,\mathbb R^m)$, i.e. every partial derivative of $f$ is continuous on $A$, and $A$ is open in $\mathbb R^n$, then $f$ is differentiable on $A$. Is the ...
4
votes
1answer
93 views

Additive norm $||a+b||=||a||+||b||$

I've read somewhere that there exist spaces where $||a+b||=||a||+||b||$ is true iff $a = \lambda b, \ \ \lambda>0$. Could you tell me what spaces have that property and what spaces don't? ...
0
votes
1answer
2k views

The definition of locally Lipschitz

I am given this definition: A function $f:A\subset\mathbb R^n\to\mathbb R^m$ is locally Lipschitz if for each $x_0\in A$, there exist constants $M>0$ and $\delta_0 >0$ such that ...
1
vote
2answers
131 views

If a sequence of summable sequences converges to a sequence, then that sequence is summable.

Let $(a_i)^n$ be a sequence of complex sequences each of which are summable (they converge). Then if they have a limit, the limit sequence $(b_i)$ is also summable. All under the sup norm for ...
1
vote
1answer
35 views

In a normed space, characterization of the sets $F$ such that $d(x,F)$ is achieved for all $x$?

Let $(E,\|.\|)$ be a real normed vector space and $d$ be the distance associated to the norm. I am wondering if there exists a characterization of the subsets $F$ of $E$ such that for all $x\in E$, ...
0
votes
1answer
352 views

Is this valid: Every Cauchy sequence in a normed space is absolutely convergent.

Proof. Let $X$ be a normed space with norm $|\cdot |$ and $(x_n)$ be Cauchy. Then for all $\epsilon \gt 0, \ \exists N : m,n \gt N \implies |x_m - x_n| \lt \epsilon$ is the standard definition of ...
1
vote
0answers
60 views

Codimension in X, given the codimension in a subspace of X

Let $X$ be a normed space with infinite dimension, let $M$ be a dense subspace of $X$ with codimension $k$ and let $N$ be a dense subspace of $M$ with codimension $l$ in $M$. The codimension of $N$ ...
0
votes
1answer
90 views

Trouble in proving that $\|x\|_p = \max|x_j|$

We define p-norm in this way: $\|x\|_p = \{\sum ^N_j_=_1|x_j|^p\}^ {1\over p}$ We know that It change to $\|x\|_p = \max|x_j| $ when $ p \to \infty $ How can I prove this ?
1
vote
1answer
394 views

Is every normed vector space a metric space?

I was trying to prove that every normed space is a metric space, and the first three proprierties came natural. However, when faced with proving the triangle inequality I had a bit of problems. I ...
1
vote
1answer
56 views

existence of a weakly cauchy sequence if the dual space is separable [closed]

Let X be a normed space such that $X^*$ is separable. Given any sequence $(x_n)\in X$ then there exist a subsequence weakly cauchy
0
votes
1answer
84 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.
1
vote
1answer
171 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
vote
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
107 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
votes
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
vote
1answer
108 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
votes
2answers
416 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?
0
votes
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 $B$. ...
0
votes
1answer
111 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 ...
0
votes
0answers
38 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
63 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 ...
1
vote
1answer
266 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
votes
1answer
76 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
63 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
123 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 ...
2
votes
0answers
234 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
votes
1answer
95 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
vote
1answer
62 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
196 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
75 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
195 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
413 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
vote
1answer
90 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 ...
0
votes
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 ...
0
votes
0answers
62 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 ...
1
vote
2answers
111 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 ...
1
vote
0answers
133 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
votes
1answer
215 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
74 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
vote
1answer
131 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 ...