# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

**5**

votes

**3**answers

216 views

### Is the Fell-Doran problem trivial in a topological setting?

The Fell-Doran problem is a problem in functional analysis. It goes as follows: Let $A$ be a complex unital algebra, $X$ a locally convex space, and $L(X)$ the algebra of all continuous endomorphisms ...

**3**

votes

**4**answers

788 views

### Is there a use for a Hilbert space that uses a different norm than the one induced by the inner product?

$l_1$ minimization / compressed sensing comes to mind. Does anyone have any concrete examples? Or is such a construct completely useless?

**6**

votes

**2**answers

800 views

### Closed, complemented subspaces of $l^1(X)$ when $X$ is uncountable

... are all isomorphic to $l^1$ on some other index set. At least, that much I "know" from 2nd-hand sources, since the original proof is apparently in a paper of Köthe from the 1930s 1960s (in ...

**52**

votes

**13**answers

3k views

### Notions of convergence not corresponding to topologies

This question concerns the ramifications of the following interesting problem that
appeared on Ed Nelson's final exam on Functional Analysis some years ago:
Exam question: Is there a metric on the ...

**37**

votes

**7**answers

5k views

### What's an example of a space that needs the Hahn-Banach Theorem?

The Hahn-Banach theorem is rightly seen as one of the Big Theorems in functional analysis. Indeed, it can be said to be where functional analysis really starts. But as it's one of those "there ...

**14**

votes

**2**answers

3k views

### Basis of l^infinity

Is it possible to exhibit a (Hamel) basis for the vector space l^infinity, given by the bounded sequences of real numbers?

**5**

votes

**1**answer

1k views

### Are smooth functions on an uncountable sum continuous?

Consider the linear space $\sum_{\mathbb{R}} \mathbb{R}$. As in the Frolicher-Kriegl-Michor view, we make this into a Frolicher space as follows.
Equip it with the locally convex topology of the ...

**15**

votes

**4**answers

1k views

### Can one do without Riesz Representation?

In more detail, can one establish that the continuous linear dual of a Hilbert space is again a Hilbert space without appealing to the Riesz Representation Theorem?
For me, the Riesz Representation ...

**3**

votes

**0**answers

368 views

### Neglect of Compact Quantum Metric Spaces [closed]

Does anyone have an opinion on Rieffel's theory of compact quantum metric spaces? To me it seems to be a very interesting new area of mathematics. It shows how to generalise complicated geometric ...

**13**

votes

**1**answer

805 views

### Gelfand-Naimark from the category-theoretic point of view

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative C* algebra (with unit) A and the C* algebra of continuous complex-valued functions on its ...

**1**

vote

**3**answers

1k views

### Various Cartan's Lemmata

I am a bit amazed by "Cartan's Lemma".. I have so far seen it in :
Algebraic Geometry sources:
Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...

**5**

votes

**3**answers

585 views

### Regularity of sparse Fourier transforms

Suppose $F$ has discrete Fourier transform $(a_n)$ where $a_n=0$ unless $n=2^k$ for some $k > 0$, in which case $a_n=1/k$ (or $a_n=1/k^2$ if you want: I'm happy with anything polynomial). What ...

**9**

votes

**2**answers

3k views

### Where was/is Compensated Compactness used?

This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...

**18**

votes

**3**answers

1k views

### Is there an L^p tauberian theorem?

From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that ...

**2**

votes

**3**answers

2k views

### What function has fourier series the harmonic series?

I know that this is on the boundaries of what's allowed, but hopefully someone'll answer before it gets closed!
What (periodic) function has Fourier series the harmonic series? I really want the ...

**3**

votes

**3**answers

1k views

### Error analysis of implicit functions

I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms):
$$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...

**1**

vote

**1**answer

795 views

### Decoupling lemma for the Lambda(p) problem

I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...

**3**

votes

**3**answers

1k views

### Minimizing a functional

I have wondered the problem in http://www.helsinki.fi/~hmkokko/Stuff/Esdale/index.html for over year without success. If we try to minimize the functional equation
T(\theta ) = \int_0^L\frac {dx}{v_0\...

**4**

votes

**3**answers

404 views

### uniformity for Banach algebras - a three space property?

Let $A$ be a commutative, unital Banach algebra and $I \subset A$ an ideal such that $I$ with the relative norm is a uniform Banach algebra and $A / I$ with the quotient norm is uniform as well.
Does ...

**23**

votes

**2**answers

2k views

### When is a locally convex topological vector space normal or paracompact?

All locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is ...

**21**

votes

**4**answers

3k views

### Are proper linear subspaces of Banach spaces always meager?

Let X be a Banach space, and let Y be a proper non-meager linear subspace of X. If Y is not dense in X, then it is easy to see that the closure of Y has empty interior, contradicting Y being non-...

**6**

votes

**5**answers

907 views

### What is a rigorous statement for “linear time-invariant systems can be represented as convolutions”?

In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of ...

**10**

votes

**2**answers

745 views

### A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...

**8**

votes

**5**answers

758 views

### Abelianization of GL(H)

This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups.
I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-...

**8**

votes

**1**answer

550 views

### opposite Banach space

I heard this from Haskell Rosenthal many years ago.
If V is a complex vector space, say the opposite of V is the complex vector space with the same elements, the same operations except switch scalar ...

**11**

votes

**3**answers

498 views

### Making an l_2 distance out of l_1 distance

If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell.
Making the grid finer doesn't ...

**20**

votes

**2**answers

1k views

### In a Banach algebra, do ab and ba have almost the same exponential spectrum?

Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...

**2**

votes

**1**answer

1k views

### Hilbert Space as direct sum of subspaces with cyclic vectors

Ok,so this should be easy, however I havent taken functional analysis for a while. But given a compact self-adjoint operator on a hilbert space H(over the complex numbers), we define v to be a cyclic ...

**8**

votes

**1**answer

318 views

### Is there a coalgebraic characterisation of the hyperfinite II_1 factor?

Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the ...

**2**

votes

**1**answer

888 views

### Range of a Certain Linear Operator

Consider the following hermitian form on the sobolev space H^1(I), of an interval I:
g(u,v):= \int_I (du/dt dv/dt - \rho(t) u v)dt, where \rho is a nice bounded function on I.
Riesz representation ...

**25**

votes

**5**answers

5k views

### Is there a natural measures on the space of measurable functions?

Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...

**19**

votes

**5**answers

3k views

### Isomorphisms of Banach Spaces

Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...

**7**

votes

**3**answers

668 views

### L_p norm balls for 1<p<2 - is it always similar to an L_q norm ball for some q>2?

The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same ...

**1**

vote

**1**answer

281 views

### Convergence of Affine Transformations

Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:
...

**10**

votes

**4**answers

709 views

### Can you describe the image of the exponential map B(H)->B(H).

James Tener asks at the 20-questions seminar:
The exponential map exp:B(H)->B(H) is just defined by its Taylor series. Can you describe its image?