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.
2
votes
1answer
17 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
35 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 ...
1
vote
0answers
28 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 ...
-1
votes
0answers
44 views
normed vector space [duplicate]
Prove that $\|x\|=\left(\displaystyle\sum_{k\in\mathbb{N}}∣x_k∣^p\right)^{1/p}$ is not norm for $\ell_p=\left\{x=(x_k)_{k\in\mathbb{N}}:\displaystyle\sum_{k\in\mathbb{N}}∣x_k∣^p < \infty\right\}$ ...
0
votes
1answer
66 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
22 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
79 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
42 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
46 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 ...
13
votes
4answers
212 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
45 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
35 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
28 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
38 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:
...
1
vote
0answers
68 views
Inner Product and Norm Defined Over General Vector Space?
Consider a vector space $\mathcal{V}$ over a field $\mathcal{F}$. The inner product is defined as a mapping
$$\langle\bullet, \bullet\rangle:\mathcal{V}\times\mathcal{V}\rightarrow\mathcal{F}$$
...
0
votes
1answer
33 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
47 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
46 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
27 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
32 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 ...
8
votes
1answer
75 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
40 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
32 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
50 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
21 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 $E$ ...
2
votes
1answer
60 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
43 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 ...
1
vote
2answers
30 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
36 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
$$\|f\| = \|f\|_\infty + \sup_{x,y\in ...
0
votes
1answer
47 views
Showing a norm preserving isomorphism of vector spaces
Lets define $l^1$ as the complex vector space of all absolutely summable sequences of complex numbers and and $x_0$ consists of all the sequences in $l^\infty$ (all bounded sequences) that eventually ...
1
vote
2answers
53 views
how to show that $A_kB_k\to AB?$
Let in the space $M(n,\mathbb R)=$ set of all $n\times n$ real matrices endowned with $\| \cdot \|_2,~A_k\to A,~B_k\to B.$ Then how to show that $A_kB_k\to AB?$
2
votes
0answers
56 views
Distance from a point to a plane in normed spaces
How can we calculate the distance from a point to a plane
in normed spaces ? where we are not inner product, for example:
Calculate the distance in $(C[0,1],||\cdot||)$ endowed with the supremum norm ...
2
votes
1answer
62 views
How to proof homeomorphism between open ball and normic space
How can I prove that an open ball $B$ in a normed vector space $X$ is homeomorphic to $X$?
3
votes
1answer
56 views
Prove the boundedness of a bilinear continuous mapping.
Let $X,Y,Z$ are Banach spaces and $$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.$$ Is ...
1
vote
0answers
32 views
Definition of a projection on a normed space? Banach space?
Given a vector space $V$, a projection $V\to V$ is an idempotent linear map.
For a normed space do we require anything else of the definition like continuity? Is the image required to be closed in ...
1
vote
1answer
26 views
Distance of a function from a subspace
Let $f \in L^2([-a,a])$. Trying to find $\mathrm{dist}(f,S)$ in $L^2([-a,a])$ (where S is the subspace of real polynomials of max degree $2$, like $a+bx+cx^2$) and knowing that $\langle f,a\rangle=0$ ...
0
votes
1answer
22 views
Determinant of Schur Complement
If I have an $n \times n$ real-valued non-symmetric matrix $\mathbf{M}$, which has determinant $|\mathbf{M}| > 0$, what can I say about the determinant of the matrix $\mathbf{Q}^T \mathbf{M}^{-1} ...
1
vote
2answers
63 views
Is't a correct observation that No norm on $B[0,1]$ can be found to make $C[0,1]$ open in it?
There's a problem in my text which reads as:
Show that $C[0,1]$ is not an open subset of $(B[0,1],\|.\|_\infty).$
I've already shown in a previous example that for any open subspace $Y$ of a ...
3
votes
1answer
95 views
Weakly compact implies bounded in norm [duplicate]
The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous.
If a subset $C$ of $X$ is compact for the weak topology, then ...
2
votes
1answer
51 views
Spectrum of linear operators
I can't solve the following:
i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$
Find $\sigma(T)$.
ii) Let $S : l^2 ...
2
votes
2answers
43 views
If $X$ is a normed space and $Y \subset X$, show $\max\limits_{\substack{f \in X^*,\\ \|f\|\leq 1,\,f|_{Y}=0\;}} |f(x)|=\inf\limits_{y \in Y}|x-y|$
Let $Y \subset X$ a subspace of normed space $X$. Show that
$$\displaystyle \max_{f \in X^*, \ ||f||\leq 1, \ f|_{Y}=0} |f(x)|=\inf_{y \in Y}|x-y|.$$
4
votes
1answer
35 views
Let $(X, \|.\|)$ be an NLS, $x\in X$ and $0 < r<s.$ Show that $B(x, r)\subsetneq B (x, s).$
In my text I've found the problem:
Let $(X, \|.\|)$ be an normed linear space, $x\in X$ and $0 < r<s.$ Show that $B(x, r)\subsetneq B (x, s).$
I can see if $\exists~y\in X-\{0\}$ then ...
3
votes
1answer
43 views
Comparing norms on $\mathbb{R}^n$
We know that $\mathbb{R}^n $ is normed linear space with respect to the norms defined as follows
$\Vert x\Vert_{1} = \sum_{i =1}^n |x_i|$
$\Vert x\Vert_{2} = (\sum_{i =1}^n |x_i|^2)^{1/2 }$
$\Vert ...
0
votes
1answer
39 views
Questions regarding Holder's and Minkowski's inequality
I've some questions regarding Holder's and Minkowski's inequality as given in my text:
Does the author consider the case $q=\infty$ in the equality case of lemma 1.1.36?
Shouldn't the author ...
1
vote
1answer
143 views
Question over function twice differentiable if $D^2 f$ is constant
Let $E$ and $F$ be normed spaces. What can you say of a function $f:A\subseteq E\to F$ with $A$ open in $E$ twice differentiable, if $D^2 f$ is constant?
This is a very open question that do not ...
0
votes
1answer
43 views
Bounded Derivate of a differentiable and Lipschitz function
Let $E, F$ normed spaces and $f:A\subseteq E\to F$ with $A$ open set, suppose that $f$ is differentiable at $a\in A$ and that $f$ is locally Lipschitz of constant $k>0$ in $a$. Show that ...
8
votes
3answers
180 views
Does $\|\cdot\|_2:C_\mathbb R([0,1],\mathbb C)\to\mathbb R:f\mapsto\sqrt{\int_0^1|f(t)|^2dt}$ come from any inner product?
I'm trying to show $\|\cdot\|_2$ is a norm on the $\mathbb C$-vector space $C([0,1],\mathbb C)$ where $$\|\cdot\|_2:C([0,1],\mathbb C)\to\mathbb R:f\mapsto\sqrt{\int_0^1|f(t)|^2dt}$$
I've stuck in ...
0
votes
1answer
77 views
Exercise of differentiable functions in $\mathcal{C}[0,1]$
Consider $E=\mathcal{C}[0,1]$ with norm $\|\cdot\|_\infty$. For which $x$ is differentiable the following functions:
a) $f:E\rightarrow E$ defined by $f(x)(t)=|x(t)|^{2/3}$
b) $f:E\rightarrow ...
1
vote
1answer
32 views
Special operator on a normed space
Let $E$ be a normed space and $T \in L(E)$ with $\|Tx\|\lt\|x\|$ for all $x\ne0$ and $\|T\|=1$.
I want to prove the following:
$A=\{x\in E: \|Tx\|\ge1\}$ is closed.
There is no $x\in A$ with ...
1
vote
1answer
52 views
Differentiability of the supremum norm in $\ell^{\infty}$
Let $\ell^{\infty}=\{x\in \mathbb{R}^{\mathbb{N}}: x\,\, \text{is bounded}\}$ and $E=\{x\in \ell^{\infty}:x_n\rightarrow 0\}$ with the norm $||\cdot||_{\infty}$ and let $f(x)=||x||_{\infty}$. How to ...
