All Questions
Tagged with metric-spaces linear-algebra
196
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3
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79
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Completeness meaning (complete basis vs complete metric space)
Today my professor started talking about the formalism of QM.
We talked about the eigenvectors of a Hermitian operator (over Hilbert space) as a "complete set". He also mentioned briefly ...
- 389
-1
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1
answer
64
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How do I prove the symmetry of the Hyperbolic metric on the Half-Plane?
Okay so the preamble I'm working with is that in the upper half-plane model of hyperbolic space, we are equipped with an inner product defined on a vector $v$ originating at the point $z=x+iy$ as: $\...
- 9
2
votes
1
answer
120
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Shortest distance between two affine subspaces through orthogonal projection
I'm trying to show the following:
Let $V$ be a finite dimensional euclidean vector space, with two vector subspaces $S_1 \subset V$ and $S_2 \subset V$. Suppose that $X, Y$ are affine subspaces with $...
- 33
1
vote
1
answer
49
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Universal metric of a bivariant tensor $t^i_j$.
I'm reading Elements for Physics by Albert Tarantola, and get stuck on page 16.
Before this page, the author defined a metric on a linear space $S$ as a map to its dual $\mathbf{G} : S \to S^*$ that ...
- 1,111
2
votes
2
answers
124
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How to calculate the distance from $X^2$ to $\operatorname{span}(1,X)$?
Question:
Let $E = \Bbb{R}_2[X]$ an euclidian space with a dot product $\left\langle P,Q\right\rangle \ = \int^1_0 {P(t)Q(t)dt}$.
Calculate the distance from $X^2$ to $\operatorname{span}(1,X)$.
...
- 133
1
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0
answers
58
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Parameterizing matrices whose rows and columns sum to a constant
I have a constrained optimization problem for a matrix $A \in \mathbb{R}^{m \times n}$. The objective doesn't matter, and the constraint is that
for some scalars $\alpha, \beta \in \mathbb{R}$, we ...
- 836
2
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0
answers
94
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Metrizing pointwise convergence of *sequences* of functionals in a dual space.
Let $X$ be a normed, real vector space of uncountable dimension. Let $X^*$ denote the set of all continuous linear functions from $X$ to $\mathbb{R}$. Does there exist a metric $d : X^* \times X^* \to ...
0
votes
1
answer
139
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Geometric Interpretation of the Gram-Matrix
I wanted to ask what the geometric interpretation of the Gram-Matrix is. In our linear algebra course, we defined it as:
$$G(\vec{v_1}..\vec{v_n}) =
\begin{bmatrix}
\langle\vec{v_1},\vec{v_1}\rangle &...
user1066956
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0
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35
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Completetion of Banach spaces for uncountable index
Let $I$ be an index set, and $V_i$ is a collection of Banach spaces. Consider $p < \infty$, I know that $\sum V_i$ is may not complete under the norm $$||f|| := \left(\sum ||\pi_i(f)||^p\right)^{1/...
- 629
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2
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50
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open sphere around $x=14$ with radius $r=3$ for closed set $[10,16] \cup [17,25]$, is $16$ an interior point?
Let $ S=[10,16] \cup [17,25] \subseteq \mathbb{R}$ and
$ d: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R} $ with $ \mathbb{R}: d(x, y)=|x-y| $.
I need to find the open sphere around $x=14$ ...
1
vote
2
answers
163
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prove that "closed and bounded" implies "compact" in a finite dimensional normed space
I'm trying to prove that "closed and bounded" is equivalent to "compact" in a finite dimensional normed space using the following lemma:
Let $x_1,...,x_n$ be linealy independent ...
- 573
4
votes
1
answer
152
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$(U\cap W)^\perp = U^\perp + W^\perp$ in metric vector spaces
I'm currently working my way through Roman Advanced Linear Algebra chapter 11 and am getting caught up on 2(b).
Let $U,W$ be subspaces of a metric vector space.
Prove that $(U\cap W)^\perp = U^\perp +...
- 63
1
vote
2
answers
125
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Properties of Norm in R^2
I have a question regarding properties of norm in $R^2$:
Does $Norm(x,y) = Norm(y,x)$ in general?
If $a, b, c ≥ 0$ and we have $a≥b$, how to prove $Norm(a,c) ≥ Norm(b,c)$?
These seems to be very ...
- 103
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3
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113
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Is the metric space c0 of sequences of real numbers converging to zero is isometric to the metric space $\ell^1(\mathbb{N})$
Is the metric space c0 of sequences of real numbers converging to zero is isometric to the metric space $\ell^1(\mathbb{N})$
To begin this proof do I need to show that:
c0 is complete in $\ell^1(\...
- 205
1
vote
1
answer
115
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Hilbert space has always an orthonormal Schauder basis + what is meant by the dimension of a Hilbert space?
I realized that I have become a bit rusty when it comes to the definition of a basis and have the following question:
Usually, a basis in a Hilbert space, denoted as $H$, is defined as an orthonormal ...
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