Tagged Questions
3
votes
1answer
58 views
Can a closed (non-trivial) subspace of an incomplete vector space be complete?
While thinking about the statement:
A subspace of a complete vector space is closed if and only if it's complete.
I was trying to drop the first "complete" and see what gets broken.
And my ...
0
votes
1answer
36 views
Is $l^{\infty}$ subpace $s(\mathbb{F})$?
Let be $s(\mathbb{F})$ a all sequence with term in $\mathbb{F}$ ($\mathbb{C}$ or $\mathbb{R}$). Denoted one element of $s(\mathbb{F})$, that $x=(x_j)$. Let be $$l^{\infty} = \{x\in s(\mathbb{F}); ...
2
votes
1answer
77 views
How to prove this claim?
For $x=(x_1,\dots,x_p)$ define two norms $\|x\|_1=|x_1|+\cdots+|x_p|$ and $\|x\|=\big(\sum_{i=1}^{p}x_{i}^{2}\big)^{\frac{1}{2} }$. Find the largest constant $a>0$ and the smallest constant ...
1
vote
2answers
42 views
Need help understanding the concept of the Jacobian Matrix and its relation to differentiation
As the question suggests, I need help understanding the concepts around the following differentiation of vector-valued functions:
1) The Norm. I understand that Norm to be defined as follows:
In the ...
2
votes
1answer
56 views
Convex cone question.
Hoi, let $V$ be finite dimensional real vector space with inner product $\left\langle . \right\rangle$ and let $\Gamma \neq \{0\}$ be a closed convex cone. Let $$\Gamma_0^{\perp}:=\{v\in ...
0
votes
1answer
49 views
Pointed Convex cone: one-to-one correspondence extreme rays - extreme points
Hoi, let $V$ be a finite dimensional real vector space with inner product $\left\langle .\right\rangle$. Let $\Gamma\subset V$ and $\Gamma \neq \left\{0\right\}$ a pointed convex cone. (Pointed means ...
0
votes
1answer
67 views
Problem related to differential of a map
I dont understand how to solve this problem. Please can you explain the solution clearly? I want to learn how to solve such problems. Thank you
0
votes
2answers
128 views
what is $ M^{\perp}$ given set?
Let $ X=C[-1,1]$ be inner product space with definition $$\langle f,g\rangle =\int_{-1}^1 f \overline{g} dt .$$
Let $M$ be the subspace defined by
$$ M= \left\{f \in X\mid ...
4
votes
2answers
110 views
Is an infinite linearly independent subset of $\Bbb R$ dense?
Suppose $(a_n)$ is a real sequence and $A:=\{a_n \mid n\in \Bbb N \}$ has an infinite linearly independent subset (with respect to field $\Bbb Q$). Is $A$ dense in $\Bbb R?$
2
votes
1answer
95 views
Prove that a given subspace of $C[-1,1]$ with $L^2$ norm is closed
Let $H= C[-1,1]$ with $L^2$ norm and consider $G=\{f \in H \mid f(1) = 0\}$. Show that $G$ is a closed subspace of $H$.
I've been trying to prove this for a while but i can't establish that given ...
0
votes
0answers
28 views
Find an inner product on $C[0,1]$ such that {$2^{1/2}\sin(n\pi x):n\ge 1$} is an orthonormal set
Find an inner product on $C[0,1]$ such that {$2^{1/2}\sin(n\pi x):n\ge 1$} is an orthonormal set and verify that this set is orthonormal for your choice of inner product.
I'm pretty much stumped for ...
2
votes
1answer
166 views
Prove that a compact subset of a normed vector space is complete
Prove that a compact subset of a normed vector space is complete .
My thought process:
Suppose S is a compact subset of $(V,\parallel . \parallel)$. Since S is compact, it is closed and bounded, so ...
1
vote
0answers
43 views
Strictly convex absolutly 1 homogeneous function
Is every strictly convex, 1-homogeneous function on $\mathbb R^d$ simply a multiple of the Euclidean norm?
Update: The above is no, since any p-norm on $\mathbb R^d$ is strictly convex and ...
0
votes
0answers
66 views
Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$, then $L$ is Lipschitz.
Let $U$ and $V$ be normed linear spaces over $\mathbb{R}$, and $L : U \mapsto V$
a linear function. Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$,
then $L$ is Lipschitz.
There ...
1
vote
2answers
36 views
$\ell_{p}$ space closed to addition
I'm trying to show that $\ell_{p}$
is a vector space for any $1\leqslant p<\infty$
. So given two infinite series $\left(x_{n}\right)_{n=1}^{\infty}$
and $\left(y_{n}\right)_{n=1}^{\infty}$
...
4
votes
1answer
93 views
Differentiation continuous iff domain is finite dimensional
Let $A\subset C([0,1])$ a closed linear subspace with respect to the usual supremum norm satisfying $A\subset C^1([0,1])$. Is $D\colon A\rightarrow C([0,1]), \ f\rightarrow f'$ continuous iff $A$ ...
0
votes
2answers
67 views
Intersection of $|z_1 - x|=r$ and $|z_2 - y|=r$
Let $x,y \in \mathbb{R}^k$ ($k\geq 3$), $|x-y|=d>0$ and $r>0$.
Prove that if $2r>d$, then there are infinitely many $z\in \mathbb{R}^k$ such that $|z-x|=|z-y|=r$.
Here's what I have ...
3
votes
2answers
25 views
Boundedness of Surfaces in $\mathbb R^3$
GIven an equation such as $ax^2+by^2+cz^2+dxy+exz+fyz=g$ where $a,b,c,d,e,f,g\in \mathbb R$, How can we tell if the surface described is a bounded one without explicitly plotting a graph?
7
votes
1answer
748 views
Is this class of periodic functions closed under the (circular) convolution operation ? Help in proving.
I am studying the properties of a particular class of functions, and I'd appreciate some help in proving a property of that class. I started with a class of functions and made some modifications to ...
2
votes
0answers
84 views
Condition on $c$ for a contraction map
I am extending Example 2.2 on this sheet.
Suppose $f(x(s),s)$ is such that $|f(x(s),s)-f(y(s),s)|\leq K |x-y|$ for some $K>0 ---(1)$
and $x,y\in C[0,t_f]:\,\,\,t_f<\infty$
Also, let $T$ ...
0
votes
1answer
86 views
Field for bounded real sequences
On page 12 of Luenberger's Optimization by Vector Space Methods, he claims that the set of bounded real sequences is vector space with addition defined component-wise.
Ok, but what is the underlying ...
9
votes
2answers
660 views
How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
I want to prove the following theorem (no idea whether it has a name):
Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...
0
votes
0answers
141 views
Visualization of 2-dimensional function spaces
As a follow-up question to what is the norm measuring in function spaces
I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
0
votes
3answers
716 views
What is the norm measuring in function spaces
In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors.
The generalization to function spaces is quite a mental leap (at ...
1
vote
1answer
160 views
Dot Product and Orthogonal Complement
Let V be the vector space of all real-valued bounded sequences. Then for $a,b \in V$ $\langle a,b \rangle :=\sum _{n=1}^{\infty } \frac{a(n) b(n)}{n^2}$ defines a dot product. Find a subspace $U ...
