1
vote
1answer
42 views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
1
vote
1answer
49 views

Proof the that compact support is a vector space

Currently I am studying for my exam of Real Analysis, however there is one thing that I do not seem to get. Given: $$ \mathrm{Supp}(f):=\overline{\{x\in\mathbb{R}^n:f(x)\neq 0\}} $$ the support of ...
0
votes
1answer
21 views

Extenstion of Intermediate Value Theorem.

Let $f:[0,1]^{d}\longrightarrow \mathbb{R}^{d}$ with $d\geq 2$. $f$ is continuous and let $c\in (0,1)$. If we have that $f(0,...,0)<<(c,...,c)$ and $f(1,...,1)>>(c,...,c)$, is there an ...
2
votes
3answers
57 views

Let $V $be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

The Assignment: Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove: There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in ...
1
vote
0answers
23 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
0
votes
1answer
63 views

Banach's Fixed Point theorem - banach vector space

Problem 1 (Banach fixed point theorem). Let $(V, || \, \, \, ||)$ be a Banach space, $U \subset V$ a closed subset (in the sense that convergent sequences in $U$ have their limits in $U$) and $T : ...
3
votes
1answer
51 views

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$.

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$. Let consider $H=l_2$ where $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace $ ...
0
votes
1answer
40 views

Showing that some differential equation has an infinite dimensional solution space?

I don't see how to proceed or even where to start to show this thing that I have found: The differential equation $$(\sin x)\frac{dy}{dx} - 2(\cos x)y = 0$$ has an infinite solution space of ...
0
votes
2answers
50 views

$l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
1
vote
0answers
49 views

Real version of the Jensen's formula.

Prove the Jensen's formula $$\int_{T}f(z+re^{2\pi i\theta})d\theta-f(z)=\iint_{D(z,r)}\log{\frac{r}{|w-z|}}\Delta f(w)dm(w)$$ where $w$ is in $D(z,r)$ and $f$ is a two-dimensional $C^2$ ...
3
votes
1answer
74 views

Show that a normed Vector space is complete, need smart help.

I want to show that a normed vector space is complete. I know that if you can show that every Cauchy sequence converges, then it is complete. But in a normed vector space, completeness is equivavlent ...
2
votes
1answer
72 views

Where is piecewise dirichlet function with $|x|^2$ continuous or differentiable?

If $|x|^2$ is continuous and differentiable on all of $\mathbb{R}^n$ (already shown differentiability by showing all $n$ of its partial derivatives are continuous), then... Question: For the function ...
0
votes
1answer
61 views

Homework excercise, completeness in Vector-spaces, is it correct?, long, but can it be simplified?

I have a very difficult excercise. I see now that it became too much text for someone to might go through it, if you can please help me, but don't want to read all, can you please then only answer my ...
13
votes
5answers
1k views

Why is one proof for Cauchy-Schwarz inequality easy, but directly it is hard?

Let's say you are in $\mathbb{R}^n$ and you define the norm as $||x||=\sqrt{x_1^2+x_2^2...+x_n^2}$. This we recognize as the usual norm from the inner product: $||x|| = \sqrt{\langle x, x \rangle}$, ...
6
votes
2answers
136 views

Question about Normed vector space.

Here is the definition of a normed vector space my book uses: And here is a remark I do not understand: I do not understand that a sequence can converge to a vector in one norm, and not the ...
0
votes
1answer
45 views

How to prove sum of vectors with same magnitude is equal to zero.

Suppose that we have $n$ vectors $v_1,v_2...v_n$ with same magnitude in plane s.t. the angle between $v_i$ and $v_{i+1}$ is $2\pi/n$ then $v_1+v_2+...v_n=0$ for all $n \geq 2$. I can show this by ...
0
votes
0answers
32 views

linear continuation

I want to ask something about the following setting Let $F$ be a normed linear space, $E\subseteq F$ a subspace and $G$ a Banach space. Let $T: E \rightarrow G$ a bounded linear map. I have to prove ...
1
vote
1answer
39 views

Are these inner product spaces?

1) Vector space of $2\times2$ real matrices and $(A,B)=\text{trace}(AB)$ 2) Vector space consisting of all polynomials of degree $2$ with $\langle p,q\rangle=p(-1)q(-1)+p(1/2)q(1/2)+p(-1)q(-1)$ How ...
0
votes
0answers
39 views

Simple question about natural domain of a function and open/closedness of the domain

This is an easy problem, I just want to make sure I understand everything correctly. I want to find the natural domain of the function $\large\sin(\frac{1}{xy})$ and describe whether it is open or ...
1
vote
1answer
58 views

Orthogonal complement in pre-hilbert space

I just want to be sure that the following is correct: Let $T:H \rightarrow H$(continuous), where $H$ is a pre-Hilbert space, then we have $H=\ker(T) \oplus\ker(T)^{\perp}$, where $\ker(T)^{\perp}$ is ...
1
vote
1answer
82 views

Show that all $f$ integrable on $[0,1]$ with $\int_0^1 f(x)dx = 0$ is a vector space

Let $V =$ the set of $~f$s integrable on $[0,1]$ with $\int_0^1 f(x)dx = 0$. I want to show that it is closed under addition. Let $f$ and $g$ be in $V$. $f + g$ be in $V$. Then since the integral of ...
0
votes
0answers
17 views

Given data, approximations in a metric space for moving into a normed vector space isometrically.

Please see this question and this answer. Here $f_x(y)$ is approximated by $$x_v = [d(x,K_1),d(x,K_2),....d(x,K_N)]$$ by choosing to consider distances from $x$ to only certain points $K_i$ and ...
2
votes
2answers
153 views

Sub-dimensional linear subspaces of $\mathbb{R}^{n}$ have measure zero.

I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is: Given $k<n$ every $k$-dimensional subspace of $\mathbb{R}^{n}$ has ...
3
votes
0answers
41 views

Choosing an appropriate complete orthogonal basis

I have a function $f(x)$ which I want to represent as the sum over some complete orthogonal basis $\phi_i$ such that: $$ f(x) = \sum_{i} c_i \phi_i(x) $$ Where the $\phi_i$ are orthogonal with ...
4
votes
2answers
115 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
72 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 ...
3
votes
1answer
131 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
44 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
85 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
220 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
2answers
103 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
65 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
113 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
181 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
122 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
154 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
1answer
108 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
377 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
71 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
115 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
39 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
150 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
69 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
26 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
887 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
98 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
104 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
997 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
217 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
1k 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 ...