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)

3
votes
0answers
20 views

How numerical radius help us to conclude an operator is normal and partial isometry?

In Furuta's book, "Invitation to Linear Operators" there is a theorem, theorem 2 in 3.7.3, that says: If $T^k=T$ for some integer $K\ge 2$ and if $w(T)\le 1$, then $T$ is the direct sum of a unitary ...
2
votes
3answers
24 views

Can distributions be thought of as functions of a real variable?

I understand that, given some function space, distributions lie in the dual space. In that sense, they can be thought of as functions of a "function of a real variable" variable. But the common ...
2
votes
1answer
40 views

Correctness of proof that weak convergence implies pointwise convergence in C([0,1])

I want to prove that in the space of (complex-valued) continuous functions on the real interval [0,1] equipped with the sup norm, which I will denote by $\mathscr{C}([0,1])$, weak convergence implies ...
0
votes
0answers
16 views

dimension set of all fourier series functions and basis

Let $F$ be the set of all functions on $D = [0,2 \pi] \times [0,2 \pi]$ that have a convergent Fourier series. Is the dimension of $F$ equal to $\aleph_{0}$? Given a countable infinity of linearly ...
0
votes
1answer
26 views

Euclidean geometry and $L_2(\lambda)$ space

Suppose $f,g\in L_2(I,\lambda)$ with $\lambda$ any probability measure and the norm $\| x\|=\sqrt{\langle x, x\rangle}$. Could we have the same geometric properties in this space as in the Euclidean ...
0
votes
1answer
12 views

An inequality for power of positive functions

Let $f,g,h$ be positive real vlaued functions on a finite set $\mathbb{X}$. Let $p >1$. I am wondering whether the following should be true? $$\sum_{x\in ...
3
votes
0answers
30 views

some important proofs about adjoint operators

I was told that the formal adjoint of the gradient is the negative divergence. Let $A : H\to H$ be a bounded, linear operator, The adjoint of $A$, i.e. $A^*: H\to H$ satisfies \begin{equation*} ...
0
votes
2answers
21 views

Understanding Quotient spaces - Shrinking down

I am looking at Page 57 of Kreyszig's Functional Analysis, and I have been given an exercise: Let $X=\Bbb R^3$ and $Y=\{(\xi_1,0,0)| \xi_1 \in \Bbb R\}$ 1) Find $X/Y$: So $X/Y=\{[x]:x+Y,\forall ...
2
votes
1answer
18 views

The convergence of a product of sequences converging in $L^2$.

Earlier today I found myself pondering the following question for which I do not have a reasonable answer. Suppose $f_m\to f$ and $g_m\to g$ in $L^2$. Moreover suppose that $f_m g_m\in L^2$ for ...
1
vote
1answer
18 views

Poisson functional on bounded domain

I was wondering if it is actually clear that on bounded domains the Poisson integral is bounded from below: $$I[u]=\int_{\Omega} \left( \frac{1}{2}\lvert \nabla u \rvert^2 - u\rho \right)\, dx,$$ I ...
0
votes
0answers
30 views

Limiting value of $L^2$ functions

Let $f\in L^2(\Omega)$, where $\Omega \subset \mathbb{R}^2$ is the unit square $[0,1] \times [0,1]$. Let $x\in \Omega$. Suppose I evaluate $f$ at points from some direction that approach $x$. ...
3
votes
0answers
37 views

Find a function which satisfies an integral equation

How can I find a function, $$ f: \mathbb{R} \to \mathbb{R} $$ which satisfies the following equation: $$\cos\left(t^2\right) = \int_{-\infty}^{\infty} e^{itx}f(x)\,dx$$
3
votes
1answer
47 views

Elegant way to prove that the space must be infinite dimensional?

Let $F(S,V)$ be the set of all functions from S to a vector space V, assume that $V\ne\{0\}$, and that S contains infinitely many elements, then we must have that $F(S,V)$ is ...
0
votes
1answer
23 views

Radius of convergence of Taylor expansion of $z \mapsto (1 - z \cdot a)^{-1}$

Let $A$ be a Banach $\mathbb{C}$-algebra with norm $\text{N}(-)$ and let $a \in A$. Where can I find a reference to/can somebody supply a proof of the following posited equality?$$\max_{z \in ...
1
vote
0answers
20 views

The restriction of a discontinuous linear functional to any open set is surjective.

Problem. Let $X$ be a topological vector space and $f:X\to\mathbb{K}$ a linear mapping. Prove that if $f$ is discontinuous, then $f(A)=\mathbb{K}$ for all nonempty open set $A\subset X$. I'd like ...
3
votes
1answer
49 views

A simple $C_{0}$-semigroup question.

Let $u:[0,t_{e}]\to\mathcal{D}(A)$ satisfy $$\begin{cases} \frac{du}{dt}=Au & 0\le t \le t_{e} \\ u(0)=x \end{cases}$$ I want to prove that necessarily $u(t)=T(t)x$. So it's clear to see that ...
0
votes
2answers
49 views

$f(I)\cap g(J)\not=\phi$ for all open interval $I,J$

Let $f,g:\mathbb{R}\rightarrow \mathbb{R}$ and $f(I)\cap g(J)\not=\phi$ for all nonempty open interval $I,J$. Consider $f_1=\chi_\mathbb{Q}$ and $g_1=\chi_\mathbb{Q^c}$, we know that $f_1$ and $g_1$ ...
-1
votes
0answers
27 views

Is continuous random variable always an “onto function” [on hold]

Is continuous random variable always an "onto function"? If yes, why?
0
votes
1answer
23 views

Is probability mass function (PMF) the “law of X”?

Are they two the same? If not, what's the differences between these two? In continuous case, is PMF also equal to the integration of probability density function?
2
votes
1answer
43 views

Show that the spectrum of an operator on $\ell^2(\mathbb{N})$ is $\{0\}$.

The problem I have comes from Walter Rudin's Functional Analysis, chapter 10 exercise 19. The exercise begins with the following: Let $S_R$ be the right shift operator, acting on ...
0
votes
0answers
47 views

Linear Independence for functions defined by integration [on hold]

Given that the set of functions $$f_i(x,y), \quad i=1,\dots,n$$ are linearly independent for $(x,y) \in [0,1]^2$. Is the set of functions, $g_1,\ldots,g_n$, defined by $$ g_i(x) = \int_{y\in [0,1] } ...
4
votes
2answers
60 views

How to simplify $ \int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx $ using Green's indentity?

Let $\varphi\in C_c^\infty(\Bbb{R^2})$ (infinitely differentiable functions with compact support) and consider $$ I=\int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx, $$ the existence of which is ...
0
votes
0answers
15 views

The dual of the space of $p$-locally integrable functions

If $X$ is a space of finite measure, what is the dual space of $L^p _{loc}$ (the space of locally $p$-integrable functions)? When $p=1$, a good answer has already been provided. What is known for $p ...
1
vote
1answer
19 views

continuously differentiable and local contraction

Let $F$ be a map from $\mathbb{R}^n$ to $\mathbb{R}^n$. Fix $x_0\in \mathbb{R}^n$. If $F$ is continuously differentiable near $x_0$ and the spectral radius of the Jacobian of $F$ at $x_0$ is less ...
1
vote
0answers
25 views

Given a spectrum, what can we know about its function?

Say we are given a well-behaved function $f(t)$ (either discrete or continuous) and are able to compute its spectrum $F(k)$ using the (discrete) Fourier transform. Then say we lose $f(t)$ and know ...
1
vote
1answer
30 views

Negativity of Convex Combinations

Consider the functions $f(x)$, $g_1(x)$ and $g_2(x)$ with following properties: $\int f(x) dx =\int g_1(x)dx =\int g_2(x)dx =1$. Define the following measure of negativity for the functions: ...
0
votes
2answers
34 views

Weak convergency vs strong convergency in Hilbert space

Let $\mathcal{H}$ be an Hilbert space and let $(x_n)_n \subset \mathcal{H}$ be a sequence s.t. $$ x_n \rightharpoonup x ~~~,~~~ \| x_n \| \to \|x\| $$ We want to show that $ x_n \to x $. Now, I ...
1
vote
2answers
98 views

Characterization of the weak topology

In our functional analysis lecture we defined the weak topology in a what seems to me like a non canonical way, i.e. not as unions of finite intersections of preimages of open sets in the underlying ...
2
votes
2answers
20 views

A closed set A and compact set B in a topological vector space.

I don't believe this is a repeat question. I have seen it asked before on here, but not in this way. If I take a closed set $A$ in a topological vector space $X$ and a compact set $B$ also in $X$, is ...
1
vote
0answers
29 views

What powers of $|x|$ belong to $L^1$?

Prove that $|x|^ {−qp} \in L^{1}(U)$, where $U=B_{1}(0)\subset \mathbb{R}^{n}$. I think I could use polar coordinates to facilitate the work but not sure if it is useful.
2
votes
2answers
46 views

Question about vector spaces with the discrete topology

Is it true that every vector space with the discrete topology is a topological vector space? (That is, a topological space with continuous addition and scalar multiplication whose singletons are ...
-1
votes
0answers
28 views

Understanding the set structure of probability theory [on hold]

Since events have their own probabilities and outcomes have their own probabilities. Why don't we just consider only one of events or outcomes directly? What's the motivation to have this set-point ...
7
votes
2answers
104 views

Real analytic functions

I'm writing because I don't know the usefulness of real analytic functions. I mean, I know that analyticity is something more respect differentiable ($C^\infty$ function), but I don't have in mind a ...
0
votes
1answer
22 views

Exercise: Uniform Boundedness Principle and Double dual

Let $X$ be a normed vector space and $(x_{n})$ be a sequence in $X$. Show that if the sequence $f(x_{n})$ is bounded for every $f \in X^{\ast}$, then there exists $C > 0$ such that $\|x_{n}\| < ...
0
votes
0answers
54 views

Specific Root of Interpolating Polynomial

We define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is fixed $n-1$, $\beta$ is chosen uniformly at random from the field of $p$ elements. We evaluate $P$ at some $x_i$ values. So we get ...
2
votes
1answer
29 views

Do approximate identities remain approximate identities if one adjoins 1 to a C* Algebra?

If we have a C* Algebra $\mathscr{U}$ without an identity we can adjoin an identity $\mathbb{1}$ in the following way: We take $\mathscr{\tilde U}$ to be the set $\{(\alpha,A); \alpha \in ...
2
votes
1answer
23 views

Unitary elements in Banach spaces and subspaces.

Let $F$ be a Banach space and $E$ be a subspace of $F$. Let $e_{0}\in E$ be an element of norm $ 1$ and suppose that span $\{f\in F^{*}:\|f\|=f(e_{0})=1\}=F^{*}$, where $F^{*}$ is the dual space of ...
0
votes
1answer
36 views

Is there a name for operators of the type $A: M \to M$

In some theorem in functional analysis I have noticed that it is important to assume that an operator $A: M \to M$ where $M$ is some set plus conditions, as opposed to $A: M \to N, M \neq N$ Is ...
0
votes
0answers
12 views

Density of intersection of sets with boundary condition

I would like to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{n}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) : ...
2
votes
1answer
33 views

Why is the Span of a subset of a linear space defined in such at way?

If I have a subset $M$ of a linear space $E$, we define the linear span of the subset, $M$, as: $$\operatorname{span} M=\bigcap_\alpha \{E_\alpha : E_\alpha \hookrightarrow E\text{ and } M \subseteq ...
-1
votes
2answers
56 views

intuition of mass function of random variable [on hold]

When we are using $P\{X=x\}$ it seems like intuitively there is a function from $T$ (or measure from $\mathcal{B}(T)$) to $[0,1]$. What is the theoretical foundation behind this intuition?
2
votes
1answer
37 views

Why does the limit $ \lim_{\varepsilon\to 0+}\int_{\varepsilon}^M \frac{\varphi(x)-\varphi(0)}{x}\ dx$ exist for smooth $\varphi$?

Let $D(\Bbb{R}):=C_c^\infty(\Bbb{R})$ and $$ p.v.(1/x)(\varphi):=\lim_{\varepsilon\to 0}\int_{|x|>\varepsilon}\frac{\varphi(x)}{x}\ dx. $$ for $\varphi\in D(\Bbb{R})$. I'm trying to understand ...
4
votes
1answer
47 views

Example of Topological Vector Space

Is there a topological vector space such that, for every $x\in X$, there is a proper neighbourhood $V$ of $x$ in $X$ which is convex, but the whole space is not locally convex (i.e. $X$ has a local ...
0
votes
1answer
52 views

Example of Heine-Borel Theorem

For a subset $S$ of Euclidean space $\mathbb R^n$, S is closed and bounded if and only if S is compact (that is, every open cover of $S$ has a finite subcover). I need an example or ...
2
votes
1answer
28 views

Is $\ell_1$ complemented in its double dual $\ell_1^{**}$? (i.e., in $\ell_\infty^*$?)

Quick question, y'all. Is $\ell_1$ complemented in $\ell_1^{**}=\ell_\infty^*$? Yes, I searched Google, and also the standard texts. I can't seem to find an answer, but surely this is known. ...
0
votes
0answers
18 views

Show that the continuum of elements $e^{i\lambda t}$ forms a complete orthonormal subset of $B^2$.

Let $X$ be the vector space of all finite linear combinations of functions of the form $e^{i\lambda t}$ ($-\infty<t<+\infty$), where the parameter $\lambda$ is real. An inner product in ...
1
vote
1answer
39 views

parallelogramm law and inner products

Is the sum of two norms $||.||_1=\sqrt{(.,.)_1}$ and $||.||_2=\sqrt{(.,.)_2}$, where $(.,.)_1$ and $(.,.)_2$ are the quadratic forms of inner products on a normed linear space, again produced by an ...
0
votes
0answers
16 views

“Asymptotic” $\mathbb{R}$-algebras

Definition. By an asymptotic $\mathbb{R}$-algebra, I mean an $\mathbb{R}$-algebra $F$ of functions $\mathbb{R} \rightarrow \mathbb{R}$ satisfying: $$\mathop{\forall}_{f:F}\left[\left(\lim_{x ...
0
votes
0answers
18 views

What do we call the number that measures how good of an asymptote $g$ is to $f$, and what are the basic results about this number?

Suppose we have a (potentially very complicated) smooth function $f : \mathbb{R} \rightarrow \mathbb{R},$ and we're trying to approximate it (in the limit as the input value goes to $+\infty$) by a ...
2
votes
2answers
41 views

“Scalar product” of two Lp spaces

I was reading the book A. Lasotta and M. C. Mackey, "Chaos, Fractals, and Noise: Stochastic Aspects of Dynamic", Springer, 1991 On page 27, they defined a ``scalar product'' as follows. Let ...