All Questions
Tagged with vector-spaces normed-spaces
685
questions
3
votes
2
answers
114
views
Nature of the Euclidean Norm
I've been re-reading my linear algebra book and a definition is given of the norm of a vector in $\mathbb{R}^n$ to be:
||v|| $= \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$.
For $\mathbb{R}^2$ and $\mathbb{...
- 197
1
vote
0
answers
56
views
Spanning set of support functionals in dual space
I am currently studying about supporting hyperplane (or, support functional) in dual space. Since, I am new in these topics I met with the following queries:
Let $X$ be a normed space and $X^*$ be the ...
- 178
-1
votes
2
answers
79
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Why is positive definite defined this way?
Norm: A norm on a vector space $V$ is a function $\| \cdot \| : V \to \mathbb R$ which assigns each vector $x$ its length $\|x\| \in \mathbb R,$ such that for all $\lambda \in R$ and $x, y \in V$ the ...
0
votes
1
answer
33
views
$\|f^{-1} \|= \frac{1}{a} $
We define
$$B(X,Y)= \{ f \mid f:X \to Y, \text{ is continuous and linear function}\}$$
Let $X,Y$ bee Banach normed spaces, $f \in B(X,Y)$, $a>0$, and $ \|f(x)\| \ge a\|x\|$ for all $x\in X$.
Then
$...
- 159
0
votes
1
answer
33
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For normed spaces $ E=\{x\in X \mid \inf\{ \|x-e\| \mid e \in E \}=0 \} $
Let X be a normed vector space and $ E \subset X$
Prove that $$ E=\{x\in X \mid \inf\{ \|x-e\| \mid e \in E \}=0 \} $$
I tried to prove like this:
Let
$\begin{align*}
x \in E &\Rightarrow 0 \le \...
- 159
0
votes
1
answer
34
views
About strictly convex norm
Let X be a normed vector space. $\| . \|$ is a norm.
we said this norm is a strictly convex norm if $$ \forall x,y \in X : \| x\| \le 1, \|y\| \le 1 \Rightarrow \| \frac{x+y}{2} \| <1 $$
I have ...
- 159
0
votes
2
answers
76
views
Let $X$ be the vector space of all real sequences that have at most finitely many non-zero terms. Is $(X, \| \cdot \|)$ a Banach space?
Let $X$ be the vector space of all real sequences that have at most finitely many non-zero terms. I was able to show that the prescription $\| \{x_n\}_{n \in \mathbb{N}} \| = \max_{n \in \mathbb{N}} |...
user1316777
1
vote
1
answer
61
views
prove that $\|f\|=1$, where $f : X\to X/\mathscr M$ is the projection
We define
$$B(X,Y)= \{ f \mid f:X \to Y, \text{ is continuous and linear function}\}$$
Let $X$ be a normed vector space and let $\mathscr M \subset X $ be a closed subspace. Define
$$X/\mathscr M :=\{...
- 159
1
vote
1
answer
94
views
Prove that $\|f\|=n^{1/2} $
We define
$$B(X,Y)= \{ f \mid f:X \to Y, \text{ is continuous and linear function}\}$$
We said that $f$ is linear bounded function if $$ \exists M>0 : \forall x\in X, \|f(x)\| \le M\|x\| $$
Also, ...
- 159
1
vote
1
answer
23
views
Vector Norm $\vert\vert v\vert\vert_V$ expressed as supremum of $\vert Lv\vert$ over all bounded operators L with $\vert\vert L\vert\vert_{op}\leq 1$
Let $V$ be a normed vector space with norm $\vert\vert\cdot\vert\vert_V$. How can I show that for all $v\in V$ we have
$$\vert\vert v\vert\vert_V = \sup\{\vert Lv\vert \:\colon\: L\in \text{Hom}(V,\...
- 567
2
votes
0
answers
87
views
Ratio of l1/l2 norm over n dimensional vector
According to here
$$\frac{(|c_1|+|c_2|+\cdots+|c_n|)^2}{c_1^2+c_2^2+\cdots+c_n^2}\geq 1$$
The equality holds when for each $i\neq j\in[n]$, $|c_i||c_j|=0$.
Could we improve the lower bound by ...
- 1,052
0
votes
0
answers
27
views
Strict convexity of norm
Let $E$ be a Banach space and $B(E)$ (resp.~$S(E)$) be the closed unit ball (resp.~the unit sphere) of the Banach space $E$. $E$ has strictly convex norm if for each pair of elements $x, y \in S(E)$ ...
- 1,313
1
vote
0
answers
30
views
Generalised directions
Let $X$ be a normed space, $A$ a real vectorspace. Let a map $\phi:X \setminus \{0\} \to A$ such that it satisfies the following properties
$\phi(\lambda x) = \phi(x)$ for every $\lambda >0$
if $\|...
- 591
0
votes
1
answer
24
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let X is a normed vector space. if $ D \subseteq X$ is a balanced set $ D^0 \cup \{0\}$is a balanced set.
I tried to prove that:
"Let $X$ be a normed vector space. If $D \subseteq X$ is a balanced set then $D^0 \cup \{0\}$ is a balanced set."
$D^0$ is the interior of $D$.
I tried to prove it ...
- 159
1
vote
2
answers
56
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"Best" Submultiplicative / Subordinate norm?
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especially, $K$ can be a subordinate matrix norm. I ...
