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
votes
2answers
112 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 ...
2
votes
1answer
58 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
85 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 ...
1
vote
1answer
35 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 ...
1
vote
0answers
39 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
167 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
70 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
120 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
79 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 ...
1
vote
2answers
36 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. ...
1
vote
2answers
41 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
58 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 ...
0
votes
0answers
20 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
311 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
142 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
42 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
83 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
74 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) = ...
4
votes
2answers
131 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 ...
2
votes
2answers
328 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
112 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
71 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
654 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 ...
1
vote
1answer
91 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 ...
1
vote
1answer
176 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
317 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 ...
1
vote
1answer
177 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
42 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
28 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
52 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 $x$?
1
vote
2answers
190 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
148 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 ...
1
vote
1answer
46 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 $x \in X$ ?
1
vote
2answers
75 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
2answers
69 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 $n$?
17
votes
4answers
480 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
42 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
1answer
69 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 ...
2
votes
1answer
80 views

Find norm of a linear functional

I have a normed space: $$l_p = \{ (x_n)_{n = 1}^{\infty}: \sum\limits_{n = 1}^{\infty}|x_n|^{p} < \infty \}$$ With norm: $$||x|| = (\sum\limits_{n = 1}^{\infty} |x_n|^p)^{1/p}$$ where $p = ...
2
votes
1answer
49 views

Calculate $\|f\|$ with $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$

In $C[-1,1]$, consider the norm $\|x\| = \max\{x(t)\mid t \in [-1,1]\}$ and $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$, $x \in C[-1,1]$. Find $\|f\|$ Hi everybody. I got stuck in this problem. I can ...
0
votes
1answer
96 views

Show that orthogonal complement is trivial

I have this subspace of $C[-1,1]$ with inner product $\langle f,g\rangle = \int_{-1}^1f(x)\cdot \bar g(x)\,dx$: $$ E=\left\{f : \int_{-1}^0f=\int_{0}^1f\right\} $$ need to prove that $E^\bot=\{0\}$
1
vote
1answer
25 views

Closure of a set and and an “Open Ball” in a normed space

Let $X$ be a normed space and fix $t \in X$. Set $T = \{x\in X : ||x-t|| ≤ r\}$ and $S= \{x\in X : ||x-t|| < r\}$. Definition: $\mathbf{y}\in\operatorname{Closure}(T)$ if there exists a sequence ...
2
votes
1answer
127 views

Strange mistake in this proof about normed separable vector spaces.

I am supposed to prove that an infinite-dimensionale separable normed space $X$ constains a countable subset $Y$ that is linearly independent and dense. There are three hints given: First, prove ...
3
votes
2answers
91 views

$T$ be the operator from $C[0,1]$ to $C[0,1]$ defined by $Tf = f'+f''$. Show that the operator $T$ is unbounded.

$f \in C[0,1]$, the space of all continuous, complex-valued functions on $[0,1]$ with supremum norm. $\|f\|=\sup_{x\in[0,1]}|f(x)|$. Let $D$ be the set of $f \in C[0,1]$ such that the first ...
0
votes
1answer
37 views

“Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs). Then $||AB||_{op} \leq ...
0
votes
2answers
42 views

If $||f-g|| < ||f^{-1}||^{-1}$, then $f$ is isomorphism implies $g$ is also isomorphism

Let $E$, $F$ be Banach space, $f,g \in L(E,F)$ and $f$ is isomorphism. Prove that if $||f-g|| < ||f^{-1}||^{-1}$, then $g$ is isomorphism. Hi everybody. I got stuck on this problem and can't ...
1
vote
0answers
71 views

Linear subspace of normed space

Let $X$ be a normed space and $A \subset X$ such that $X \setminus A$ is a linear subspace of $X$. Show that A is either dense or empty set. Can someone check my proof? If $A$ is empty set we ...
1
vote
1answer
46 views

A normed space of continuous functions with norm $\int_{0}^{1}|f(t)|dt$ is not complete

Suppose $E$ is a normed space of all continuous functions on $[0,1]$ with norm $\int_{0}^{1}|f(t)|dt$. Prove that $E$ is not complete I know that we must do is to find a Cauchy sequence of ...
1
vote
1answer
51 views

Can $||f|| = ||a||_{q}$ to arbitrary values of $p$ and $q$ satisfying ${1 \over p} + {1 \over q} = 1$

We all know that: Suppose $a = (a_{1}, a_{2}, ..., a_{n})$ is a point in Euclide space $R^{n}$. Consider the mapping $f: R^{n} \rightarrow R$, $f(x) = \sum_{i=1}^{n}a_{i}x_{i}$. Then $||f|| = ...
1
vote
0answers
19 views

Prove that if Y is a closed subspace of a normed linear space X and x \nin Y, the span of x and Y is closed. [duplicate]

The question is stated in the title, but I don't see how to prove it. Each element of $span\{x,Y\}$ is of the form $cx + y$ where c $\in \mathbb{R}$ and $y \in Y$. If I could show that for any ...