3
votes
1answer
54 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 ...
1
vote
2answers
47 views

Hilbert vs Inner Product Space

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product. Also can a metric always be defined ...
6
votes
2answers
112 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
0answers
12 views

find medoid of a set of objects described by their non-metric distances.

suppose i have a set of objects, their coordinates are unknown to me, I am given their symmetric pairwise distances, you can imagine a matrix of size n-by-n where (i, j)th element is the distance ...
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 ...
1
vote
1answer
52 views

Is there a metric space and meanwhile a linear space such that vector addition discontinuous but scalar multiplication operation continuous?

Some special problems about topological groups or topological linear space theory. Recently I have done some study in some respects about topological group, topological linear spaces. And I found it's ...
1
vote
1answer
50 views

Is there any non-translation invariant but homogeneous metric linear space?

A metric linear space is a metric space and vector space, and linear operation is continuous regarding to the metric. I know that a homogeneous, translation invariant metric $d$ can be used to define ...
1
vote
1answer
42 views

Pairwise Maximum Metric

Well , I had question on vector spaces and maximum metric . Lets us assume a set of vectors of $N$ dimension containing only integers , and let us make a set of vectors then we will calculate the ...
0
votes
1answer
50 views

Variations in math to implement three-dimensional space?

Backstory: So I was researching topics, and found that 3-D game programming often markets itself with linear algebra. As a philosopher of math I decided to dig further into this and determine if ...
0
votes
3answers
41 views

Prove that this series converges?

I have a Banach space $X$ and a linear operator $A \in L(X)$. $A$ is bounded such that $||A|| <1$. I then have to show that $$log(I-A)=\sum_{n \ge 1} \frac {A^n}n$$ converges. All I can come up ...
1
vote
1answer
58 views

Prove this is a bounded linear operator and find its operator norm?

I have a map $$A:(C[0,1], || \centerdot ||_\infty) \rightarrow \mathbb R, Ax = x(0) \forall x \in C[0,1]$$ and need to prove it's a bounded linear operator, and find its operator norm. I've tried ...
0
votes
1answer
19 views

find examples to prove $ A \cup B $ is not part of the $ V $

For $ A, \: B $ is the subspace of $ V $ Find examples to prove $ A \cup B $ not a subspace of $ V $ I learn about this program should not know how. Desire to help people and give a solution ...
1
vote
1answer
46 views

Metric on an infinite dimensional space with equivalence relation.

In analyzing a problem I've come across a space defined by the following equivalence relation: $(\cdots, x_{-2}, x_{-1}, x_0, x_1, x_2, \cdots) \sim (\cdots, z^{-2}x_{-2}, z^{-1}x_{-1}, x_0, zx_1, ...
0
votes
1answer
126 views

Is the set closed, open, or neither?

Consider $C[0,1]$, the normed linear space of all real-valued continuous functions within the given interval. The norm endowed on this space is $\|f\|_{\infty} = \sup_{x \in [0,1]} |f(x)|$. Consider ...
-2
votes
1answer
49 views

Are all metric space as a euclidean space?

I believe that all euclidean space is a metric space? But I need to know about inverse? I mean: are all metric space as a euclidean space? Is there any kind of metric space which is not euclidean ...
3
votes
5answers
91 views

Minkowski sum of two disks

An open disk with radius $r$ centered at $\mathbf{p}$ is $D(\mathbf{p}, r)=\{\mathbf{q} \mid d(\mathbf p, \mathbf q) < r\}$, and the Minkowski sum of two sets $A$ and $B$ is $A \oplus B=\{\mathbf p ...
5
votes
2answers
143 views

why we want to use grassmannian space?

I wonder what's the special about grassmannian space? Why we want to use this space? On wikipedia, it says: "By giving a collection of subspaces of some vector space a topological structure, it is ...
2
votes
2answers
81 views

Proving the normed linear space, $V, ||a-b||$ is a metric space (Symmetry)

The following theorem is given in Metric Spaces by O'Searcoid Theorem: Suppose $V$ is a normed linear space. Then the function $d$ defined on $V \times V$ by $(a,b) \to ||a-b||$ is a metric on $V$ ...
1
vote
1answer
80 views

Is a normed topological space metrizable?

As stated in the title: If there is a norm on a topological space, then we get a metric induced by the norm. Is this true?
0
votes
3answers
113 views

Function to uniquely map a set of rectangles in space to a number?

I am trying to build a new way of indexing spatial data. Is there a function that takes as parameter a number of rectangles in euclidean space, and outputs an unique number?Can such a function be ...
1
vote
0answers
62 views

To show that something is a four-vector

I hope this question is not too inane... it would be really helpful for me to have this cleared up. I want to know what I need to show to demonstrate that something is a four-vector. I have checked ...
0
votes
1answer
72 views

Find a number that minimizes distance to a vector of sets of numbers

Assumptions $V$ is a vector of sets $V_1,V_2,...,V_n$ of numbers: $V=[V_1, V_2,..., V_n]^T, \forall_{i=1..n}V_i\subset\mathbb{R}$ $c\in\mathbb{R}$ is constant $d(V,c)$ is an error metric: ...
7
votes
2answers
103 views

What structure does the space of functions into $X$ (or the cartesian exponentiation of $X$) inherit from $X$?

When dealing with a space $X$, that posses a lot of structure (complete lattice, complete metric space, vector space), what can be said about the cartesian exponentiation $X^Y=\{f \mid f:Y\rightarrow ...
-1
votes
1answer
111 views

Practical implications of a vector space being a topological vector space

I have a space $V$ and I lately discovered that it's a topological vector space. What are the practical implications of that?
1
vote
1answer
36 views

Recommend a space to analyze the bearing of the vector between any two points

In Euclidean space, given any two points, the vector connecting them can be characterized by length (distance) and direction (bearing). Now I am only interested in the bearing part. And I found it is ...
0
votes
1answer
36 views

If $a=b+ c$ and $E$ is the basis of $a$, Will $E$ be also basis for $b$ and $c$?

Suppose $a$ lies in the span of a set of independent vectors $E$. Now, if $a=b+c$, is it also the case that $b$ and $c$ lie in the span o the same set of vectors $E$? if the question is obscure, ...
4
votes
5answers
2k views

What is the difference between metric spaces and vector spaces?

Does a metric space have an origin? That is, does it have $(0,0)$. Does a vector space have an origin? It seems whatever you can do in a metric space can also be done in a vector space. Is this ...
7
votes
2answers
2k views

Cosine similarity / distance and triangle equation

There is a similarity function particular popular for processing sparse vectors such as textual data (word frequency counts etc.) commonly referred to as cosine similarity. There are two variants to ...
1
vote
1answer
89 views

What is the proper term for the entity that relates a vector space and a set?

One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) ...
1
vote
1answer
226 views

How Many Movable Ways(vector) In Pure $N$th-Dimensional Space?

In my opinion, In pure $2$th-dimensional space, There is 2 movable ways. And in pure $3$th-dimensional space, There is 3 movable ways. Am I think in right way? Any answers will be appreciated, ...
18
votes
2answers
3k views

Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...