1
vote
2answers
65 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 ...
1
vote
1answer
30 views

Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space. Does anyone have any ...
1
vote
1answer
54 views

Superspace as the Hilbert Space for Quantum Gravity

This is a question I've asked in physics.stackexchange, but have obtained no answers: Let $\mathcal{A}$ be the Ashtekar connection. Since $^{(3)}g_{AB}=i\frac{\delta}{\delta\mathcal{A}^{AB}}$ (see R. ...
9
votes
3answers
197 views

Conjecture: the function $d(x, y):=\frac{||x-y||}{\max(||x||, ||y||)}$ is a distance

I make the following conjecture: the function $$ d(x, y):=\frac{||x-y||}{\max(||x||, ||y||)} $$ is a distance on $H$, where $H$ is a normed vector space or a Hilbert space, and $x, y \in H$ (the ...
5
votes
3answers
189 views

Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
1
vote
1answer
127 views

It is possible to generalize the “real” line to be able to embed $\omega_1$ or any uncountable ordinal into a finite segment of it?

This question is motivated from a previous question, but is in itself independent of it. So, I understand that it is not possible to embed $\omega_1$ or any uncountable ordinal into the real line, ...
0
votes
1answer
67 views

Show that $C([0,1],\mathbb{R})$ with the $L_2$ inner product norm is not a Hilbert space.

I need to prove that all continuous functions on the closed set $[0,1]$ is not a Hilbert space. Given the $L_2$ norm. I guess I need to show that every Cauchy sequence in the space, does not ...
1
vote
0answers
93 views

Reproducing Kernels are Positive Definite. Does the converse hold true?

Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...
2
votes
1answer
186 views

Notation: Representer Theorem for Reproducing kernel hilbert spaces

Am studying the basic concepts of RKHS and the representer theorem: In $f(x_i)=<f,k(x_i,\mathbb{.})>$, what does $ f$ on the r.h.s denote? What is its structure-is it a vector? I was thinking ...
1
vote
1answer
171 views

Countable Hilbert Spaces

I have seen a simple proof that no banach space over $\mathbb{R}$ can be of countably infinite dimension. However since the space of all square integrable functions on the unit interval forms a ...
1
vote
0answers
431 views

Proof of Isometry: Inner Product Preserving Map

For known points $x_i,x_j,\ldots,x_k$, in $\mathbb{R}^n$, consider a mapping $y_i,y_j,\ldots,y_k$ in $\mathbb{R}^n$ produced by minimizing the function $f(y)=\sum_{i,j} \left \langle x_i,x_j \right ...
2
votes
1answer
144 views

A question on norm of error vector

Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...