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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3
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95 views

Weak continuity in Sobolev Spaces

First consider the following two Sobolev Embedding Theorems. Theorem 1: The continuous embedding $W^{1,p}(\Omega) \subset L^{p^{*}}(\Omega)$ holds provided the exponent $p^{*}$ is defined as ...
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votes
1answer
88 views

Norm of Matrix transpose

I have a problem below: Let $\|\cdot\|$ denotes the norm matrix \begin{equation} \|A\|=\max \frac {\|Ax\|}{\|x\|}, \end{equation} for every $A$. Now suppose that $H: \mathbb{R}^k \rightarrow ...
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vote
1answer
102 views

Closure of $B=\{ f\in C'[0,1] : |f(x)|\leq 1, |f'(t)|\leq 1 \forall t\in[0,1]\}$ (NBHM $2005$)

If $A$ is the closure in $\mathcal{C}[0,1]$ of the set $B$ where $$B=\{ f\in C'[0,1] : |f(x)|\leq 1, |f'(t)|\leq 1 \forall t\in[0,1]\}$$ Then Which of the following is true? $A$ is closed. $A$ ...
0
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1answer
26 views

Bounded & Norm space [closed]

Can someone help me on this exercise ? Thanks!
6
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1answer
118 views

Isomorphisms between Normed Spaces

Between two normed (linear) spaces there are several notions of isomorphisms: Linear isomorphisms: linear bijective maps Topological isomorphisms: linear homeomorphisms (due to the linearity these ...
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vote
1answer
46 views

About the closedness of Banach space

I'm reading a proof in trying to prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive. There's a point in the proof saying that $X$ is a closed subspace of $X^{**}$, but I ...
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0answers
60 views

Linear Operators: Continuous $\Rightarrow$ Bounded

Let $T:V\rightarrow V'$ be a continuous linear operator between two normed vector spaces $V,V'$. Show that it is bounded. Continuity is defined as $\lim_{n}\|x_n-x\|=0\Rightarrow ...
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2answers
127 views

Are the two infima equal?

Let $R$ be a ring (not necessarily commutative) or an algebra over some field with a norm defined on it and let $I$ be an ideal in $R$. Let $a,b \in R$. Does it hold that $\inf_{i,j \in I}\|ab + ai ...
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3answers
92 views

If every inner product space can be converted into a norm space, then why is there a distinction between the two?

If every inner product space can be converted into a norm space, then why is there a distinction between the two? $$\|x\| = \sqrt{\langle x,x\rangle }$$
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0answers
87 views

normed space functional analysis [closed]

applying this lemma for e1=(1,0,0) e2=(0,1,0) e3=(0,0,1) what is maximum of c? i found c=1 am i right?
3
votes
2answers
90 views

Completeness/Compactness of a subset in a normed linear space

Let $(X,\|\cdot\|)$ be the normed linear space consisting of the sequences $a=(a_n)_{n=1}^{\infty}$, for which the corresponding series $\sum_{n=1}^{\infty} a_n$ converges absolutely, with norm ...
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vote
1answer
48 views

Endpoint-average inequality for a line segment in a normed space

Let $X$ be a normed vector space over $\mathbb R$. What is the smallest universal constant $C>0$ such that the inequality $$\|x\|\le C\int_0^1 \|x+tv\|\,dt\tag{1}$$ holds for all $x,v\in X$? ...
2
votes
1answer
43 views

The proof of the triangle inequality of this norm

Let $V$ be a normed vector space and $W$ a closed subspace of $V$. It is possible to define a norm on $V/W$ (the quotient space) by defining $\|v + W \|_{V/W} = \inf_{w \in W} \|v + w\|$. As an ...
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vote
1answer
28 views

Proof of uniqueness of zero for this norm

Let $V$ be a normed vector space and $W$ a closed subspace of $V$. It is possible to define a norm on $V/W$ (the quotient space) by defining $\|v + W \|_{V/W} = \inf_{w \in W} \|v + w\|$. As an ...
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0answers
28 views

Proof about scalar multiplication of this norm

Let $V$ be a normed vector space and $W$ a closed subspace of $V$. It is possible to define a norm on $V/W$ (the quotient space) by defining $\|v + W \| = \inf_{w \in W} \|v + w\|$. As an exercise I ...
4
votes
3answers
154 views

Continuous Linear Functional on $\ell^{\infty}$

I'd like help answering two questions. 1) Prove that there is a continuous linear functional on $\ell^\infty$ such that $f(e_n)=0 \ \forall n \in \Bbb{N}$ and $f(a)=5$ where $a=(1,1,1,1,1,1,\ldots)$. ...
3
votes
0answers
56 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
2
votes
2answers
90 views

Is $C[a,b]$ a closed linear subspace of $L^{p}(a,b)$

I am not sure about the last step of my proof: $(L^{p}(X,A,\mu), \|\cdot \|)$ is a normed $L^{p}$ space of p-integrable functions. $L^{p}(a,b)$ is the space of p-integrable functions on (a,b). ...
2
votes
1answer
60 views

Find the norm of $A$ where $(Af)(t)=tf(t)$

I have the following problem that I would like to ask you about: I have $X$ as my normed linear vector space and $B(X,X)=B(X)$ as my space of all operators $A: X \to X$, where for all $A \in B(X)$ is ...
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vote
2answers
30 views

Another question about integrable functions with a transform

I am an engineering student, and taking a real analysis course at demand of my advisor, my inexperience in proofs is giving me hard time. I stumbled upon this example, whose proof left as an exercise. ...
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vote
2answers
35 views

Negative exponential distance

Let $X := \left\{(a_k)_{k \in \mathbb N}, a_k \in \mathbb C\right\}$. Let $d\left( (a_k)_{k \in \mathbb N}, (b_k)_{k \in \mathbb N} \right) := e^{-u}$ with $u$ the smallest integer $k$ such that $a_k ...
2
votes
1answer
45 views

Differentiability of function defined as integral form

Let $H(t)=\int_{\Bbb R}|f(x)+tg(x)|^p\mathrm dx$ and $f,g\in L^p(\Bbb R)$. Then, how to prove that $H$ is differentiable and find its derivative? I think it's impossible to find it by ...
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votes
0answers
16 views

Limit of $L^r$ norm in lebesgue measure theory [duplicate]

Let $f\in L^r$ for some $r>0$ and $\mu (X)=1$. Then, prove that $\lim_{p\to 0}||f||_p=\exp(\int \log|f|d\mu)$. This is from chapter $L^p$ spaces, but I don't have any idea. How to make $\log$? ...
2
votes
1answer
119 views

Are functions in Lp space always bounded?

I know that functions in $L^2$ space have finite norms by definition, but are they also bounded "almost everywhere" ? So say for instance the following functions norm is finite but it is not bounded. ...
2
votes
1answer
64 views

About open mapping and closed range theorem

I'm self-learning Functional Analysis in Rudin's book and found some following statement hard to understand. Hope someone can help me clarify this. 1) $X, Y$ are Banach spaces, $T \in B(X,Y)$, let ...
3
votes
2answers
38 views

If a function f(x) has a p-norm then does it automatically have a (p-1)-norm ? or a (<p)-norm

Hi Id like to know if a function has a p norm does that mean it automatically has a norm for al llower values of p ?. What about higher values ?. I am trying to write a proof and would like to use ...
0
votes
1answer
65 views

Properties of subspaces of Normed Vector Spaces

How does it follow that a subset of a normed vector space cannot be open if it does not contain an open ball $B_{\epsilon}(0)$ where $\epsilon > 0$? I just want to confirm also that for normed ...
5
votes
1answer
67 views

Prove, that f is a linear map.

$U,V$ - Euclidean spaces $f:U \rightarrow V$ $f(0)=0$ $ \forall _{u,v \in U}:d(f(u),f(v))=d(u,v)$ Prove that $f$ is a linear map. I'm thinking about something like this: $||f(u+v)|| =d(f(u+v),0) = ...
3
votes
1answer
77 views

Is there always an injective map from a space in its dual space?

Today our teacher said that dual spaces are "big" and told us that this is a consequence by Hahn-Banach's theorem. So I was wondering whether the dual space of a space is always "bigger" or equal ...
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vote
2answers
118 views

Any two norms on finite dimensional space are equivalent

Any two norms on a finite dimensional linear space are equivalent. Suppose not, and that $||\cdot||$ is a norm such that for any other norm $||\cdot||'$ and any constant $C$, $C||x||'<||x||$ for ...
3
votes
1answer
84 views

Subspaces of a Topological Vector Spaces

I have a few questions about topological spaces which I am currently studying. First some definitions that I am using: Definition of subspace topology: Given a topological space $(X,\tau)$ and a ...
2
votes
2answers
61 views

Functional analysis, help to show a short result

The following problem is from the theory of compact operators: Suppose $X,Y$ are normed spaces and $T:X\to Y$ is linear. Show that if $T$ is compact and invertible then $\mbox{dim}(X)$ and ...
0
votes
2answers
242 views

Show these two norms are not equivalent?

I have the following two norms on $C[a,b]$ : $$||x||_1= \int_a^b |x(t)|dt$$ $$||x||_\infty = sup_{t \in [a,b]}|x(t)|$$ $\forall x \in C[a,b]$. I need to prove that these are not equivalent. They are ...
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vote
1answer
60 views

Prove this is a bounded linear operator and find its operator norm?

I have a map $$A:(C[0,1], || \centerdot ||_\infty) \rightarrow \mathbb R, Ax = x(0) \forall x \in C[0,1]$$ and need to prove it's a bounded linear operator, and find its operator norm. I've tried ...
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vote
1answer
115 views

Show C(X) is a vector space over $\mathbb R$ with the following operations?

I have a set of continuous functions, $C(X): X \rightarrow R$ on a compact metric space, and definitions of addition & multiplication: $$(f+g)(x) = f(x)+g(x)$$ $$(\lambda f)(x) = \lambda ...
0
votes
1answer
140 views

$X$ normed linear space separable $\Longleftrightarrow$ $\exists K \subset X$ compact s.t. $\overline{ \text{span}\{K\}}= X$

Let $X$ be a normed linear space. Show that $X$ is separable if and only if there is a compact subset $K$ of $X$ for which $\overline{ \text{span}\{K\}}= X$ I can't figure out how to solve this ...
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vote
1answer
101 views

Inequality for Euclidean norm

Let:| | be Euclidean norm on $\mathbb{R}^{n}$ and $b : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n}$ and $\sigma : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n\times m}$ two continuous functions. ...
0
votes
1answer
30 views

problem in functional analysis

let $A\subseteq‎ X$ that $X$ is normed space and for every $f\in X^*$, $f(A)$ is bounded. show that $A$ is bounded. I know must use banach-ishtenhous`s theorm.
0
votes
1answer
31 views

Lipschitz and derivatives

Show that if $f$ is bounded function on $E$ that belongs to $L^p_1(E)$, then it belongs to $L^p_2(E)$ for any $P_2 > P_1$. I am totally clueless on how to start. Is $f$ an element in a ...
0
votes
1answer
26 views

Determine all the $x_0$ such that $\phi : \mathbb C[X] \to \mathbb C, P \mapsto P(x_0)$ is continuous

In $\mathbb C[X]$, we consider the norm $\left\lVert P \right\rVert = \sup \left|a_i\right|$ for $P(X) = \sum_{i=1}^na_ix^i$. For all $x_0$ we consider the linear form $\phi : \mathbb C[X] \to \mathbb ...
2
votes
1answer
50 views

Existence of bounded linear operator with kernel reduced to $\{0\}$

If $X$ and $Y$ are normed spaces, why there must exist a bounded linear operator $T$ from $X$ to $Y$ such that $T(x)$ is not equal to $0$ for all non-zero $x$?
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vote
2answers
85 views

Inequality involving Lp space, Holder Space, Sobolev Space

Do we have the following inequality or some variation of this: $||u||_{L^{p}(U)} \leq ||u||_{C^{0,\gamma}(\bar{U})} \leq ||u||_{W^{1,p}(U)}$ for $n < p \leq \infty$ for $u \in W^{1,p}(U)$ where $U$ ...
2
votes
1answer
90 views

In a normed space, the sum of a Closed Operator and a Bounded Operator is a Closed Operator.

The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following lemma Let $T:\mathcal{D}(T)\to Y$ be a bounded linear operator with domain $\mathcal{D}(T)\subset ...
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vote
1answer
39 views

Closed subspace of Hilbertspace

Let $X$ be a norm closed subspace of a Hilbert space $\mathcal H$. Is it true that if $x_n \in X$ converges weakly to $x \in \mathcal H$, then also $x \in X$ ?
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vote
2answers
64 views

Prove that $d_{1}$ and $d_{2}$ induce the same topology

Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology. ...
4
votes
1answer
37 views

Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
16
votes
4answers
452 views

For which $L^p$ is $\pi=3.2$?

This is a silly question that came to mind after watching numberphile video on How Pi was nearly changed to 3.2. For which $p\in(1,+\infty)$ is the ratio of the perimeter of the $L^p$ disc in $\Bbb ...
1
vote
2answers
38 views

$X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq\varnothing$

Let $X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq \varnothing$ then show that $E$ spans $X$. I am trying it in a following way.... Let be the norm ...
0
votes
0answers
44 views

Explanation on the proof of the continuous Hardy inequality

Here there is a proof of the continuous Hardy inequality (theorem 2). I would like a explanation on the following passage. ...
0
votes
1answer
54 views

Prove that $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a norm for a complete space

Prove that the space $C^1([a,b])$ consisting of continuous functions in $[a, b]$ with the norm $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a Banach space. I can't prove the completeness of this ...