Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

learn more… | top users | synonyms (1)

0
votes
0answers
6 views

When two projections in a C*-algebra are “almost” Murray-von Neumann equivalent, they are equivalent.

Let $A$ be a C*-algebra and $p,q \in A$ be projections. Assume there is an element $a\in A$ such that $\|aa^*-p\|<\frac{1}{4}$ and $\|a^*a-p\|<\frac{1}{4}$. Then there is a partial isometry $v$ ...
1
vote
1answer
15 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
0
votes
0answers
5 views

What's the best way to recognize a shape o a function with N-points

I've many shapes with points in theirs countours, how is the best way to recognize a shape? I think the DTF is available but i don't know whether this is the optimal way. P.S. I think if i will ...
0
votes
1answer
7 views

Integration over subsets of the complex plane.

Original Problem: Let $\Omega\subset \mathbb{C}$ be an open set and let $f:\Omega\to\mathbb{C}$ be holomorphic such that $f\in L^{2}(\Omega)$. Show that if $B(z,r)$, the ball of radius $r$ ...
1
vote
0answers
13 views

Fourier Transforms of $L^1$ functions

Suppose that $f_n$ and $f$ are $L^1(\mathbb R^n)$ functions with $f_n \to f$ in $L^1$ sense. Then is it true that their Fourier transforms defined as $$ \hat f(\xi) := \int_{\mathbb R^n} ...
0
votes
1answer
44 views

Given $M\subset H$ and $\lambda$ a continuous linear functional, show there is a unique linear functional $\Lambda$ on $H$

Suppose $M$ is a closed subspace of a Hilbert space $H$ and $\lambda$ is a continuous linear functional on $M$ with $$\sup_{m\in M, m\neq 0} {|\lambda(m)|\over \|m\|}=c$$ Using Hilbert space ...
0
votes
2answers
36 views

Projection of a Hilbert space onto orthonormal subset.

Suppose that $\{e_1,e_2,...,e_n\}$ is an orthonormal set in $H$ and define $M\equiv span\{e_1,e_2,...,e_n\}$ Show that $M$ is closed and show that if $P$ is the projection of $H$ onto $M$ then ...
0
votes
1answer
11 views

Strong and weak extrema

I am confused about the "strength" of the two definitions. The definitions I use are the following: Let $y$ be a function defined on the set $M$. Neighborhood (0. order) of the function $y$ is the ...
1
vote
1answer
18 views

Continuous spectral theorem example

The spectral theorem can be explicitly expressed for an hermitian matrix by providing its eigen decomposition. In the more general case of a bounded self-adjoint operator with a continuous spectrum, ...
0
votes
1answer
19 views

Existence of minimum norm solution to linear equation $Tx =y$

Let $T: X \to Y$ be a bounded linear map between Hilbert spaces $(X, \langle \cdot , \cdot \rangle_X)$ and $(Y, \langle \cdot , \cdot \rangle_Y)$ (the Hilbert spaces may be complex or just real ...
2
votes
1answer
40 views

Is every closed ball (or open ball) in the Eucledean Space $R^n$ convex?

I am solving a problem and I need to use this fact: Every closed ball (or open) in the Eucledean Space $R^n$ convex? Hoever, I am not sure if it is true or not. Can anyone help? Thanks!
1
vote
1answer
25 views

Composition operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator $$ H_0^1 (\Omega) \to H^{-1}(\Omega), \qquad v \mapsto g(v) $$ is $\mathcal C^1$. Of course, $g(0) = 0$. I believe that $g \in ...
1
vote
2answers
50 views

Exercise 23 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 23 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 198). Any help will be much appreciated. Thank you in advance. Suppose $\{T_k\}$ is a collection of bounded ...
1
vote
1answer
16 views

Extending mappings on simple tensors

Consider the following situation: Let $H, K$ be Hilbert spaces and let $\Phi$ be some mapping defined on simple tensors in $H\otimes K$ taking values in $B(H\otimes K)$ with the property that each ...
0
votes
1answer
37 views

Inner Product in Hilbert Space

Let $H$ be a Hilbert space and $\phi_{1}, \dots, \phi_{n} \in H$ are linearly independent vectors. How can we construct the inner product on $H$ such that $\phi_{1}, \dots, \phi_{n}$ become orthogonal ...
0
votes
0answers
28 views

Compactness criterion

I have this compactness criterion and I want to apply it, but I don't know what I must write to see if (a) is satisfied and also for (c)? For a subset $H\subset\mathcal{BC}(\mathbb{R},Y)$ to be ...
1
vote
0answers
25 views

weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then T is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B \in ...
1
vote
1answer
17 views

Cardinality of $l_{\infty}$ is c

I need to show that the cardinality of $l_{\infty}$ is $c$, the cardinality of the continuum, where $l_{\infty}$ is the space of all bounded real sequences. Any hint is appreciated. Thanks in ...
1
vote
0answers
36 views

Continuity of a linear map on Banach space.

A function $f$ on $X$ is said to separate points of $X$ if for $x_1\neq x_2$ then $f(x_1)\neq f(x_2)$. Suppose $A$ is a subset of $Y^{*}$, the space of functionals on a Banach space $Y$, which ...
0
votes
0answers
18 views

Proving a set is a vector space, proving a norm, and that the set with the norm is a Banach space

Let $c$ be the space of sequences of real numbers that converge. That is $x\in c$ means that $x=(x_1,x_2,...)$ and $lim_{j\to \infty} x_j$ exists. It is easy to verify that $c$ is a vector space. For ...
2
votes
1answer
28 views

If any two norms on a vector space are equivalent then the space is finite-dimensional [duplicate]

I need to prove: If any two norms on a vector space are equivalent then the space is finite-dimensional. I am aware of the converse of this result that on a finite dimensional vector space any two ...
0
votes
1answer
34 views

Appoximation of Lipschitz functions by $C^1-$functions

I came across the following statements in a math book without proof. Denote $M_k$ as the set of functions from $C[a,b]$ that is K-Lipschitz continous. i.e $\forall x,y,|f(x)-f(y)|\le K|x-y|$ 1) The ...
1
vote
1answer
32 views

What does this symbol mean?

I am studying a book on functional analysis and came to a definition that started like this: Let $M$ be a set and $F: M \hookleftarrow$ a function (...) My only question is: what does the ...
2
votes
1answer
40 views

Dimension for a closed subspace of $C[0,1]$.

Let $X \subset C^1[0,1]$ be a closed subspace of $C[0,1]$ (with sup norm). Prove that $X$ has to be finite-dimensional.
1
vote
1answer
28 views

Relationship between Characteristic Function and Eigenfunction

In probability we talk about "characteristic functions" of random variables, usually written as $\Phi_X(t)=E[e^{itX}]$. Is the characteristic function in some sense an "eigenfunction" (a function f ...
3
votes
0answers
33 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
0
votes
1answer
18 views

The continous of a function in the Sobolev class

Let $f\in S$ with $S= \left\{ {f:\mathbb{R} \to \left[ {0, + \infty } \right):\int_{ - \infty }^{+\infty} {{{\left| {\hat f\left( t \right)} \right|}^2}{{\left( {1 + {{\left| t \right|}^2}} ...
3
votes
1answer
38 views

Closure of $C_0^{\infty}$ in $W^{k,p}(\Omega)$

Why is it that in the definition of $W_0^{k,p}(\Omega)$ for $\Omega$ with boundary smooth enough, we only have $D^{\alpha}u$ for all $0\leq|\alpha|\leq k-1$ vanishing at the boundary and not ...
0
votes
0answers
12 views

Dense invariant domain stable under resolvent?

I have thought about the following problem: Let $A_1\dots A_n$ a family of (unbounded) essentially selfadjoint operators on some Hilbert space $\mathcal{H}$ and $\Phi\subset\mathcal{H}$ the maximal ...
1
vote
0answers
20 views

Norm of a linear function on $c_{0}$

Let $c_0$ be the space of all sequences which converge to $0$. Let $f: c_0 \to\mathbb{R}$ by $$f(x) = \sum_{n=1}^\infty \frac{x_n}{n^2}$$ What is the value of $\|f\|$?. I am just getting used to the ...
1
vote
1answer
27 views

What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
1
vote
3answers
48 views

Basis of eigenfunctions in Banach space

I have a question about proving the existence of a basis of eigenfunctions. Assume we have a compact operator $L:\mathcal{B}\rightarrow\mathcal{B}$, where $\mathcal{B}$ is a Banach space of analytic ...
2
votes
1answer
53 views

Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
1
vote
0answers
45 views

Prove that $C[a,b]$ with inner product $\langle f,g\rangle:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space.

Prove that $C[a,b]$ with inner product $<f,g>:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space. Now the norm induced by the inner product is \begin{align} ...
1
vote
1answer
17 views

Proving that the 1-norm, $||x||_1$ is not generated by inner products on $\mathbb{C}^n$

Proving that the 1-norm, $||x||_1$ is not generated by inner products on $\mathbb{C}^n$. Is it sufficient to take $x=(1,0)$, $y=(0,1)$ in $\mathbb{C}^2$ and just showing that \begin{align} ...
1
vote
1answer
33 views

Are there any interesting Hilbert spaces that do not present as function spaces?

I was pondering this question in class earlier: All separable, infinite dimensional Hilbert spaces are isometrically isomorphic. Thus, in particular, any such space is isometrically isomorphic to ...
0
votes
0answers
20 views

How Many Negative Eigenvalues of $-\frac{d^{2}}{dx^{2}}$ on $[0,L]$?

What is the maximum number of eigenvalues $\lambda < 0$ for the trigonometric problems?: $$ \begin{array}{c} -\frac{d^{2}f}{dx^{2}}=\lambda f,\\ ...
0
votes
1answer
28 views

Ask for a good reference for the calculus involving singular continuous measure

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here. Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...
1
vote
0answers
36 views

How to select a strongly convergent subsequence from a weak convergent sequence in $L^2$?

Let $(p_n)_{n \in \mathbb N}$ be a sequence of probability density functions, which satisfies i)$$ \partial_t p_n (t,x) = \partial_{xx} (a_n (t,x) p_n (t,x)), $$ where $a_n$ is a sequence upper ...
1
vote
1answer
27 views

Two theorems about approximation by smooth functions

Let $U$ be an open subset of $\mathbb{R}^{n}$. The following are two theorems taken from the chapter about Sobolev Spaces of the Evans' book. Theorem 1 Assume $u\in W^{k,p}(U)$ for some $1\le ...
2
votes
2answers
76 views

Exercise 34 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 34 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 201): Let K be a Hilbert-Schmidt kernel which is real and symmetric. Then, as we saw, the operator $T$ whose ...
2
votes
0answers
46 views

Commuting nets for commuting projections

Let $A$ be a $C$*-algebra and $p,q\in A^{**}$ be commuting projections. Then there exist self-adjoint nets $(x_i)_i$ and $(y_j)_j$ in $A$ with $x_i\to p$ and $y_j\to q$ in the weak *-topology. Can ...
1
vote
0answers
23 views

Some kind of fundamental lemma

I remember having read something like Let $X,Y \subset \mathbb{R}^n$ open and connected and let $u \in L^1_{loc}(X\times Y)$ with $$ \int_X \int_Y u \, \mathrm{div}_y \, \varphi \, dydx = 0$$ ...
1
vote
0answers
34 views

Every Banach space is quotient of $\ell_1(I)$

I'm looking for a book containing the proof that for every Banach space E there is an index I so that E is a quotient space of $\ell_1(I)$. If I can't find the book on google books, it would be great ...
2
votes
2answers
35 views

Definition of continuous spectrum of a bounded operator

Let $T$ be a bounded operator acting on a Banach space $X$. The point spectrum $\sigma_p(T)$ is of $T$ is defined to be $$\sigma_p(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has nonempty kernel}\}$$ ...
3
votes
0answers
43 views

Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
0
votes
1answer
16 views

Dual operator of an isometry

If $X,Y$ are Banach spaces and $\phi:X\to Y$ is an isometry, show that $\phi^*$ is surjective. I can use the equality $^\perp(ran \phi^*) = \ker\phi=\{0\}$, and also use the fact that $ran \phi^*$ ...
0
votes
1answer
14 views

Continuity of operators defined via inner products.

Let $H$ be an (in general infinite dimensional) separable Hilbert space with scalar product $<\cdot,\cdot>$. Given another inner product $<\cdot,\cdot>_2$ defined everywhere on $H \times ...
0
votes
0answers
10 views

Showing $\sup_{\|b\|=1} \|ab\| = \sup_{\|b\|=1} \|b^\ast ab\|$

I wanted to show $\displaystyle \sup_{\|b\|=1} \|ab\| = \sup_{\|b\|=1} \|b^\ast ab\|$. (I also showed $\|a\|=\sup_{\|b\|=1}\|ab\|$.) One direction was easy: For all $a,b \in A$: $$ \|b^\ast a b \| ...
0
votes
3answers
42 views

motivation of definition of semigroup

I knew the definition of a group and semigroup. However, I do not see the point why we need the definition of semigroups without "good" algebraic properties as groups. Can someone motivate the ...