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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4
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2answers
143 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
59 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
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1answer
270 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
89 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? ...
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1answer
1k 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 ...
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2answers
108 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 ...
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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 $x\in E$, ...
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1answer
282 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 ...
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0answers
57 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
78 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 ?
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1answer
254 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
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1answer
50 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
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1answer
83 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
166 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 ...
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1answer
77 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
105 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 ...
4
votes
2answers
104 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 ...
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1answer
100 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 ...
3
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2answers
344 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 $B$. ...
0
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1answer
102 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
37 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
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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 ...
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1answer
207 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
71 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 ...
2
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1answer
54 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
116 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
183 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
93 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 < ...
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1answer
59 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 ...
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1answer
176 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
74 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
171 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
392 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
84 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
39 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
51 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
98 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
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1answer
36 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
129 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
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1answer
52 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 $\ker T=\{\text{ all constants }\}$ $2.P[0,1]$ ...
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0answers
38 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 ...
9
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1answer
182 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
43 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
71 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 ...
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1answer
113 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 ...