# Tagged Questions

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 ...
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}); ...
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 ...
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 ...
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 ...
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 ...
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
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 ...
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 $\Bbb R?$
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\left(y_{n}\right)_{n=1}^{\infty}$ ...
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 $A$ ...
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 ...
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?
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 ...
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 $T$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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