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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2answers
71 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
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2answers
48 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
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4answers
469 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
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2answers
41 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 ...
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0answers
54 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
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1answer
65 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
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1answer
65 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
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1answer
73 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
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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
120 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 ...
2
votes
2answers
80 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
34 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
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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
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0answers
70 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
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1answer
42 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
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1answer
45 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|| = ...
0
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0answers
329 views

When is an injective linear operator bounded below?

Given an injective linear operator $T$, densely-defined as a map between two normed linear spaces $V \rightarrow W$, what is a necessary and sufficient condition to enforce on $V$ to ensure the ...
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0answers
18 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 ...
3
votes
2answers
69 views

If $\mu(X)=1$, then $\lim_{p\rightarrow 0}\|f\|_p=\exp\left(\int_X\ln(|f|)d\mu\right)$

I am trying to prove the following result: Let $(X,\mu)$ be a measure space of measure 1 and $f$ a complex-valued function on $X$ such that there exists $r>0$ satisfying $\|f\|_r<\infty$. ...
2
votes
1answer
70 views

Infinite line is closed in $\mathbb{R}^n$

I have been reading the book "Elements of the functional analisys", by Kolmogorov and Fomin. At the chapter of Normed Linear Spaces, page 73 to be precise, the author makes the following definitions: ...
1
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1answer
51 views

a compact operator on $l^2$ defined by an infinite matrix

Let $A$ be an infinite matrix such that $\displaystyle \sum_{i,j}|a_{i,j}|^2<\infty$. Then $A$ defined a compact operator on $l^2$.
0
votes
2answers
107 views

prove that this operator is not compact

Let $g\in C[0,1]$ be a continuous function and $g\ne 0$. Let $G:C[0,1]\to C[0,1]$ the operator defined by: $G(f)(x)=f(x)g(x)$. I proved that the operator is linear and continuous. I want to prove that ...
2
votes
3answers
278 views

finite dimensional range implies compact operator

Let $X,Y$ be normed spaces over $\mathbb C$. A linear map $T\colon X\to Y$ is compact if $T$ carries bounded sets into relatively compact sets (i.e sets with compact closure). Equivalently if $x_n\in ...
0
votes
0answers
71 views

Help for applying Hahn Banach theorem in this exercise .

I want to use this version of Hahn Banach: $X$ real vector space, $M$ a subspace, $p$ a sub-linear mapping and $T:M\rightarrow\mathbb{R}$ linear and $T(x)\leq p(x)$ $\forall x \in M$ Then ...
2
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0answers
254 views

Prove: Completion of a normed space is a Banach space.

Me again with another (most likely) easy problem that is too hard for me as of now. Suppose we have a normed space $(X,||\cdot||)$ and a map $i:X\rightarrow \hat X$ with $\overline{i(X)} = \hat ...
2
votes
1answer
37 views

construction of normed linear spaces from inner product spaces

How do i construct a normed linear space from a inner product space and verify that what i have suggested is true, that is, a test of verification. Moreover, does the norm have to satisfy the ...
5
votes
0answers
120 views

Operator norm of a convolution

Consider the operator on $L^2(\Bbb R)$, $f\rightarrow f*g$, where $g\geq 0$ is some $L^1$ function. Show the operator is a bounded linear operator with operator norm equal to $||g||_1$. Showing ...
0
votes
1answer
126 views

Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional

Let $E$ be a Banach space and let $S$ and $T$ be closed subspaces, with dim$\space T<\infty$. Prove that $S+T$ is closed. To prove that $S+T$ is closed I have to show that for any limit point $x$ ...
0
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0answers
40 views

proving that some transpose operator of the transpose operator is not injective.

Let $X,Y$ be normed spaces over $\mathbb k$ (where $\mathbb k$ could be $=\mathbb C$ or $\mathbb R$). Let $T\in L(X,Y) $ (a continuous and linear operator) then we define $T':Y'\to X'$ by $T'y'=y'T$. ...
1
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1answer
99 views

Calculate the norm of this operator

$C[0,1]=\{ f : [0,1]\to [0,1], f$ continuous$\}$ $||f||_\infty=\max_{t\in [0,1]} |f(t)|$ $T:C[0,1]\to C[0,1]$ defined by $$(Tf)(t)=\int_0^1e^{s+t}f(s)ds$$ Find $||T||$ The usual way to do this ...
2
votes
1answer
134 views

Functional analysis, help in Hahn-Banach theorem application

$M$ is the subspace of $L_p[a,b]$ that $\forall f\in L_p[a,b]$ $\exists g\in M$ with $f(t)\leq g(t)$ almost everywhere. $T:M\rightarrow \mathbb{R}$ $\quad$$T(f)\geq0$ whennever $f(t)\geq0$ a.e ...
0
votes
0answers
24 views

Norm of the maximum

Consider the norm $||f||= max_{x\in[a,b]} |f(x)|$ defined in the bectorial space $C[a,b]$ I have to what is the meaning (/interpretation) in $R$ of {$||f_n-f||$}$\to$ 0 Could you help me?
1
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1answer
35 views

Let V be a normed vector space. Show that its norm is induced by a scalar product if and only if it satisfies the parallelogramm inequality.

I managed to prove left to right, but I found it hard to get the other direction even if I have the polarization identity. I've found Fréchet - Von Neumann - Jordan theorem that proves exactly what ...
3
votes
0answers
138 views

Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
1
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1answer
26 views

the set of points that are annhilated by a subset $N\subset X'$

Let $X$ be a normed vector space over $\mathbb k$ ($\mathbb k=\mathbb R$ or $\mathbb C$). Let's consider $X'$ the set of continuous linear functionals $f:X\to \mathbb k$ called the dual of $X$. We ...
5
votes
1answer
375 views

Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
0
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2answers
53 views

normed space little exercise

$(X,K)$ a normed space $E$ is a subspace of $X$. if $\exists x_0 \in X$ that $||x_0||=d(x_0,E)=1$ then show $$||e+\lambda x_0||\geq\frac{||e||}{2}\quad \forall e\in E \quad \forall\lambda\in K $$
0
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2answers
94 views

Norm space, linear operator exercise, help please!

$f \in L_2[a,b]$ $Uf(s):=\int_a^bk(s,t)f(t)dt$ $k(s,t):[a,b]^2\to R $ continuous. show 1) $U:L_2[a,b]\to L_2[a,b]$, in other words, $Uf(s)\in L_2[a,b] \quad \forall f$ 2) $U$ is linear and ...
0
votes
1answer
59 views

Show that the sequence converges to 0 under any norm in the space (R,‖.‖) [closed]

Show that the sequence $a_n = 1/n^2$ converges to 0 under any norm in the space $(\mathbb{R},\left\| \cdot \right\|)$.
2
votes
2answers
445 views

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$. I have tried to use Holder's inequality, but I still cannot solve this problem. Could you help me with this problem? Thank you ...
0
votes
1answer
46 views

Help in proving that a vector norm satisfies an axiom.

I am trying to prove if the following is a vector norm: ||x|| = max{$|x_1 + x_2|, |x_2 + x_3|, |x_3 + x_1$|} (x is vector with 3 elements) I'm stuck proving that $||\alpha x||=|\alpha|*||x||$. I ...
-3
votes
1answer
69 views

Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...
0
votes
1answer
147 views

prove that norm 1 of a function is less than or equal to (b-a) infinity norm of that function where X=C[a,b]

Prove that norm 1 of a function is less than or equal to (b-a) infinity norm of that function where X=C[a,b] is the continuous space function f:[a,b] to R. norm 1 is defined to be ntegral of the ...
0
votes
1answer
19 views

how to check wheather the given normed sequence is finite or not

Does the sequence $$ (\frac{1}{\sqrt 3}, \frac{1}{\sqrt 8},\ldots,\frac{1}{\sqrt{n^2-1}},...) $$ have finite $\ell^2$ norm? I have tried this problem using the definition of $\ell^2$ space. Is it ...
9
votes
1answer
94 views

Inner product space, points cannot be placed inside a ball of a given radius

I've found a very nice problem and I don't know how to go about solving it. Let $(E, || \cdot ||)$ be an inner product space, $x_1, ..., x_n \in E$. Prove that if for $i \neq j$ we have ...
1
vote
2answers
79 views

Continuous iff composition with every linear functional is continuous

Let $X$ and $Y$ be normed spaces, $T:X \rightarrow Y$ a linear operator. Show that $T$ is continuous if $y' \circ T$ is continuous for every $y' \in Y'$. My idea is the following: Suppose $T$ is not ...
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||} \\ ...