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)

1
vote
1answer
19 views

Selfadjoint Operators: Weak Convergence

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Regard a sequence: ...
0
votes
1answer
35 views

Laplace transform, Bochner integral

I have a quesition about linear operators on a Banach space. Let $B$ be a real Banach space. $(T_{t})_{t>0}$ is called strongly continuous contraction semigroup on $B$ if For all $t>0$, ...
0
votes
0answers
18 views

Integral kernels of self-adjoint operators

If the integral kernel $k(x, y)$ of an operator $T : C^\infty_c(M) \to \mathcal{D}'(M)$ is symmetric ($M$ is a compact manifold), then the operator $T$ is symmetric. Is the converse true? That is, ...
1
vote
1answer
53 views

A copy of $l_\infty$ in a infinite dimensional Banach space

Let $E$ an infinite dimensional Banach space. Using the Hahn-Banach extension theorem, prove that there is a sequence $(y_n)\subset E$ and a decreasing sequence of closed subspaces ...
1
vote
0answers
66 views

Existing complete function space allowing discontinuity .

This is a question which came to me due to several previous question: sorry for the all previous links necessary to look to get the question. The latest question is in the link: Convergence on Norm ...
1
vote
0answers
40 views

What's the maximum speed of snake so that the frog can escape?

Suppose there's a round pond, a frog which can swim as 1 meter / second, and a snake that moves along the pond ridge but cannot swim. If the frog can reach any point on the ridge of the pond before ...
0
votes
1answer
38 views

conditions for norm of linear bounded operator to satisfy $\lvert T_x (y) \rvert = \lVert T_x \rVert$.

Let $x = (x_n)_{n \in \mathbb N} \in l^\infty$ and let $T_x : l^1 \rightarrow \mathbb F$ be defined by $T_x (y) = \sum_{n=1}^\infty x_ny_n$. What condition on $x$ is needed so that there exists $y \in ...
4
votes
0answers
73 views

Convergence on Norm vector space.

I am not sure if this question make sense mathematically, so please bear with my ignorance. This is an extension to the question in the link: Is complete metric space is required? It seems in many ...
3
votes
2answers
35 views

Show $ \langle Tx,x \rangle \in \mathbb R$ for all $x \in H$ implies $T$ is self-adjoint

Show that a linear operator $T: H \rightarrow H$ is self adjoint if and only if $\langle Tx, x \rangle \in \mathbb R$ for all $x \in H$. You may use that the equality that for all $x,y \in H$ ...
1
vote
1answer
64 views

How do I show convergence of this sequence?

Given a sequence $\{a_n\}$ of positive real numbers such that $\sum\limits_{n=1}^\infty a_n<\infty$. Suppose that there exists $k\in \Bbb N$ such that $a_{n+k}\leq a_n, \,\,\,\forall n.$ Question: ...
9
votes
1answer
174 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
1
vote
1answer
43 views

Is complete metric space is required?

This question may be quite related to the following link: but I am not sure. Sorry, if it is trivial. Advantage/disadvantage of complete/incomplete metric space. In many application specially in ...
1
vote
0answers
32 views

Advantage/disadvantage of complete/incomplete metric space.

It must be simple. I understand a metric space can be complete for a given metric and and the same set may be incomplete with a different metric. This may be due to the fact that under the given ...
0
votes
0answers
20 views

Necessity of continuity in Topological Vector Space

In the notion of a topological vector space, we define such as a vector space $X$ (over a field $\mathbb{K}$) with topology $\mathscr{T}$ such that $$\iota_+: (X,\mathscr{T}) \times (X, \mathscr{T}) ...
2
votes
0answers
22 views

Trick to rewrite operator in terms of another?

In the book Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems by Bill Sutherland, I would like to understand the trick done in (4), see the excerpt from p 29 shown below I ...
0
votes
1answer
27 views

Bounded linear functionals over smooth maps of a compact interval

I have two questions regarding the topological dual of the space $E = \mathcal{C}^\infty([0; 1])$ of infinitely continuously differentiable functions over the closed interval $[0; 1]$ equipped with ...
6
votes
0answers
71 views

What is the strongest possible statement of the idea that “the tangent line is the best linear approximation”?

For instance, I've just checked that that if you take the best linear approximation (in the $L^2$ sense) to a sufficiently nice function $f$ on the interval $[-\varepsilon, \varepsilon]$, and then let ...
1
vote
1answer
21 views

Net converging weak-*, implies uniform bound?

Let $E$ be a complex Banach space. A consequence of the uniform boundedness principle is the following. If $(\lambda_n)_{n\geq 1}$, $\lambda$, are elements of $E^*$ such that $$ \lambda_n(x) ...
0
votes
0answers
22 views

Is there a pseudocontractive mapping that is not strictly pseudocontractive?

Given a Hilbert space $H$, a mapping $T:H\rightarrow H$ is said to be pseudocontractive if $$\|Tx-Ty\|^2\leq \|x-y\|^2+\|(x-Tx)-(y-Ty)\|^2\,\,\, \forall x,y\in H,$$ and it is strictly ...
1
vote
1answer
21 views

Show that a subset of $(\mathbb R^n,||.||)$ is closed

Let $C$ be a closed subspace of the normed linear space $(\mathbb R^n,\| \cdot \|)$.Let $r(>0)\in \mathbb R$ Define $D:=\{y:\exists x\in C$ such that $\|x-y\|=r\}$. Show that $D$ is closed. My ...
2
votes
1answer
56 views

Showing this function is continuous $ f:(x,y)\mapsto x^2+y^2$

I have the following function: $$f:\Bbb R^2 \to \Bbb R,\quad f:(x,y)\mapsto x^2+y^2$$ I want to show that this function is continuous by showing that $f^{-1}((a,b))$ is an open set. How do I ...
4
votes
1answer
56 views

How to do it by Dominated Conversgence Theorem?

I'm trying to find the limit $$ I = \lim_{n\to\infty} \int_{\mathbb R^d} \frac1{n} |f(x)|^2 x\cdot\nabla\chi (x/n)dx, $$ where $f \in H^1 (\mathbb R^d, \mathbb C)$, $f \in H^2_{loc}(\mathbb R^d, ...
0
votes
1answer
43 views

Speed as a function

we were studing the rate of the function $\frac{f{x_1}-f{x_2}}{x_1-x_2}$ if it is positive so the fonction is growing if it is negative so the function is ascending . in this moment our teacher ...
2
votes
1answer
20 views

When the singular inner part disappear in inner outer factorization?

I saw this remark in Hoffman's book - "Banach space of analytic function". If $f$ is analytic in a neighborhood of $\bar{\mathbb{D}}$, the closure of $\mathbb{D}$; then in the inner-outer ...
0
votes
0answers
28 views

Choice of a dense subset of a separable Banach space

I recently came across the following statement, and still can't prove it: Statement: Suppose $X$ is a separable,closed subspace of $L^1(G)$, where $G$ is a locally compact group. Since $X$ is ...
4
votes
1answer
75 views

Proving $\ell^p$ is complete

Let be $1\leq p\in\mathbb{R}$, denote: $$\ell^p(\mathbb {R})=\left\{(x_n)\subset \mathbb{R}: (x_n) \mbox{ is a sequence with } \displaystyle\sum_{n=1}^{\infty}|x_n|^p<\infty \right\}$$ ...
5
votes
1answer
105 views

Density of $C^\infty(\mathbb{R}^n)$ in $C^0(\mathbb{R}^n)$

This could be well-known, but I cannot come up with a rigorous proof. I want to prove density of $C^\infty(\mathbb{R}^n)$ in the continuous functions $C^0(\mathbb{R}^n)$ in the following sense: given ...
2
votes
0answers
44 views

Sequences and reflexivity

Assume X to be a real reflexive Banach space. Why are sequential topological notions topological notions ? (relatively to the weak topology on X and the weak star topology on X*) For ex : sequentially ...
0
votes
0answers
18 views

Convergence in the Implicit function theorem?

Possibly a dumb question. In the implicit function theorem we take $F\in C^k(\Lambda\times U, Y)$ with $k\geq 1$, $Y$ is a Banach space, and $\Lambda, U$ are open subsets of Banach spaces $T,X$. If ...
6
votes
2answers
70 views

Looking for a “job description” for Hölder's inequality

Here's an example of what I mean by "job description" in the post's title: triangle inequality: to be used, whenever the (unsigned) distances between adjacent points in a sequence $x_0, x_1, x_2, ...
2
votes
2answers
26 views

A criterion for invertibility of a bounded linear operator.

I'm studying Semigroup Theory and I wasn't able to understand a step in a certain proof. As far as I have been able to understand, the author used the following result: If $A$ is a bounded linear ...
0
votes
2answers
50 views

Show that $c_0$ is a Banach space with the norm $\rVert \cdot \lVert_\infty$

Let $ c_0 = \{ x = \{x_n\}_{n \in \mathbb N} \in l^\infty : lim_{n \rightarrow \infty} x_n = 0\}$. Show that $c_0$ is a Banach space with the norm $\rVert \cdot \lVert_\infty$ I am capable of showing ...
2
votes
1answer
23 views

Equivalence of sequence spaces

Let $m$ be the space of infinite sequences $(x_i), |x_i| \leq 1$ with norm $\sup_{i>0}|x_i|$. Let $\ell$ be the space of infinite sequences $(x_i), \sum_{i> 0}|x_i| \leq 1$ with norm $\sum_{i ...
0
votes
0answers
17 views

How to analyse the bound of the sum of permutation sequences?

suppose $X=[x_1, x_2, \ldots,x_n]$ ($0<x_1\leq x_2\leq \ldots\leq x_n$), and $$f(X) = \frac{x_1+2x_2+3x_3+\ldots+nx_n}{nx_1+(n-1)x_2+(n-2)x_3+\ldots+x_n}$$ i.e.,$$f(X) = ...
1
vote
1answer
14 views

Open, convex set of TVS

I'm studying LCS using Conway's book. And I had a question about a proof of Proposition 3.2 in chapter 4. The author said, the proof of this proposition is similar to that of proposition 1.14 (If V ...
9
votes
4answers
100 views

What's so special about $p=2$ for the $L^p$ spaces?

The Banach space dual of $L^p$ is $L^q$, where $q=\frac{p}{p-1}$, but I don't really understand the motivation behind this. In particular, I find it kind of surprising that the only $L^p$ space whose ...
1
vote
4answers
43 views

Monotonicity of $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x >0$.

There is this function I encountered when I was solving a problem and I am trying to study its monotonicity. So the function is $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x ...
1
vote
0answers
46 views

Is it possible to compare Sobolev space and Polish space?

Is it very easy to say that Sobolev space and Polish space are unrelated? Or we can infer some connection or relation or common structure or generalize one to another? Any comment would be highly ...
2
votes
1answer
23 views

A question on Isometry between the orthogonal subspaces of Hilbert spaces

I was reviewing my class-notes on Functional analysis and the professor had mentioned that given a closed proper subspace $U$ of an hilbert space $\mathcal{H}$, $\exists $ a closed subspace ...
3
votes
1answer
49 views

Is Riesz measure an extension of product measure?

Suppose $X$ and $Y$ are compact Hausdorff spaces and $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ are finite regular Borel measure spaces. (By regular I mean that every measurable set can be ...
-2
votes
1answer
67 views

Let we have the following exercise [closed]

How can I solve the following exercise Let $T:C[0,1]\rightarrow C[0,1]$ be defined by $$Tx(t)=y(t)=\int_0^t x(\tau)d\tau.$$ Find $\mathscr R(T)$ and $T^{-1}:\mathscr R(T)\rightarrow C[0,1]$. Is ...
2
votes
1answer
31 views

spectral theory of Laplacian on $\mathbb R^n$ [duplicate]

Can you describe the spectrum of the Laplacian $ \Delta : H^2(\mathbb R^n) \subset L^2(\mathbb R^n) \rightarrow L^2(\mathbb R^n)$? I am interested for which values $z \in \mathbb C$ the equation ...
3
votes
2answers
37 views

What is the range of the operator $T$ I mean I want to determine $R(T)$

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
-2
votes
2answers
85 views

I want some help in functional analysis [closed]

I want sone help in functional analysis : $1)$ consider the vector space $X$ of all real -valued functions which are defined on $R$ and have derivatives of all orders everywhere on $R$ define ...
1
vote
0answers
47 views

Does (L) sets in a dual Banach space X* are weak* precompact? weak* sequentially precompact?

Let $X$ be a Banach space. A subset $B$ of the dual $X$ is said to be $(L)$ set if any weakly null sequence $(x_n)\in X$ converges uniformly to zero on $B$. It is well Known in the theory that ...
1
vote
1answer
40 views

Equation in Hilbert space

Solving the following exercise of a list I have: "$H$ is a complex Hilbert space admitting an orthonormal basis $\{e_n\}, n\in \mathbb{N}$ ; $\{\lambda_n\}\subset \mathbb{C}\setminus \{0\}$ is a ...
0
votes
1answer
23 views

Is the normalized duality mapping symmetric?

In the area of functional analysis, nonlinear operator theory, the normalized duality mapping on a Banach space $X$ is a set valued map from $J:X\rightarrow 2^{X^*}$ given by \begin{align} ...
6
votes
1answer
80 views

$\sin$ and $\cos$ are the basis of what space?

When learning Fourier expansions, we learn that $\{\sin(mx), \cos(mx)\}_{m \in \Bbb N}$ is an orthonormal basis for our space and thus we can expand functions in it. My question is what space is this ...
4
votes
2answers
118 views

Are there any differences between distributions (generalized functions) and probability distributions?

A distribution/generalized function is an element of the dual space of $$S=\{f\in C^{\infty}(\mathbb{R})\colon \|f\|_{\alpha,\beta}<\infty \text{ for all } \alpha ,\beta\}$$ Where ...
4
votes
1answer
51 views

Fourier transform not surjective using oppen mapping theorem.

I know that it is possible to prove that the Fourier transform $\displaystyle\mathcal{F}: (L^1(\mathbb R),\|\cdot\|_1) \to (\{f\in C(\mathbb R): \lim_{|x|\to\infty} f(x) = 0\}, \|\cdot\|_\infty)$ is ...