Questions tagged [function-spaces]

Questions about spaces of functions, such as continuous functions between topological spaces or certain reproducing kernel Hilbert spaces. Does not concern equivalent classes of functions such as $L^p$ spaces.

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69 views

Convergence of sequence of functions in $\Bbb R^{\Bbb N}$

I'm learning about convergence of sequences in the space $\Bbb R^{\Bbb N}$ and there are couple of confusing examples. First one I have is that if $f_n \in \Bbb R^{\Bbb N}$ is given by $$\begin{align} ...
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0answers
27 views

Examples of incomplete normed spaces of continuous linear maps between two normed spaces [duplicate]

Let $X,Y$ be normed spaces, $B(X,Y)$ the normed space of continuous linear maps between $X$ and $Y$. Are there normed spaces $X,Y$ such that $B(X,Y)$ is not complete? It is well-known that $B(X,Y)$ is ...
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1answer
108 views

When is the compact-open topology discrete?

I tried to find a necessary-and-sufficient condition for the function space $\mathcal{C}(X,Y)$ equipped with the compact-open topology to be discrete. Here's my effort so far: Theorem 1. If $X$ is ...
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1answer
41 views

A mathematical representation of a circular list

An infinite list of elements of $S$ can be viewed as a function from the set $\mathbb N$ of all natural numbers to $S$. By the way, in the field of computer science, a kind of infinite list, called a ...
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0answers
28 views

Derivative of a convolution is the convolution of the derivative

In my function spaces course I was asked to prove that given if $f\in C^k_c(\mathbb{R^d})$ and $g\in L^1_{loc}(\mathbb{R^d})$ then $f*g\in C^k(\mathbb{R^d})$ and $\partial^\alpha(f*g)(x)=((\partial^\...
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16 views

Products in Besov spaces

Consider the Besov spaces $B^s_{p,q}(\mathbb R^n)$ (I am thinking about the Fourier analytic definition given in, e.g., “Theory of function spaces” by Hans Triebel). I know that $$B^0_{3,1}(\mathbb R^...
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1answer
45 views

Notation of function spaces - vector vs dimension?

I have a problem with the notation of function spaces. I append the example where my understanding of the functions space is presented. Please correct me if I did any mistake (and confirm if I did ...
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0answers
43 views

Can BMO$^{-1}$ be understood as the dual of BMO?

An standard way of characterizing elements of the function space BMO($\mathbb{R}^d$) is by means of Carleson measures. Namely we can consider the heat kernel $\Phi_t(x)=\pi^{-d/2}t^{-d}e^{-|x|^2/t}$ ...
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3answers
157 views

Are $\sin x$, $\sin(x + \frac{\pi}{6})$, and $\sin(x + \frac{\pi}{3})$ linearly independent?

I'm specifically interested in the dimensionality of the subspace spanned by $\{\sin x, \sin(x + \frac{\pi}{6}), \sin(x + \frac{\pi}{3})\}$ in $C^0(\mathbb{R}, \mathbb{R})$, the vector space of ...
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65 views

Complete regularity of function spaces

Let $X$ and $Y$ be topological spaces. Wikipedia states that if $Y$ is $T_{3½}$, then so is $Y^X$. Yet I couldn't find the proof anywhere. So here's my own attempt of a proof: (I also had had a hard ...
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44 views

Reference for interpolation theory

In the book "Interpolation theory, function spaces, differential operators" by Hans Triebel, I tried to understand the result Theorem~1(a) of section~2.4.2. In particular, I am interested in ...
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67 views

What do these symbols $C^1_0(\mathbb{R}^n)$ and $C^1(\mathbb{R}^n)$ mean?

The symbols $C^1_0(\mathbb{R}^n)$ and $C^1(\mathbb{R}^n)$ appear in this context, but I don't know the meaning of them. Let $f$ be integrable on $\mathbb{R}^n$ and $g\in C^1_0(\mathbb{R}^n).$ Then, ...
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1answer
95 views

Functional gradient and inner product in machine learning

In several applications, such as machine learning, the following setup arises: Let $V$ denote a function space equipped with the following inner product $ \langle f, g\rangle = \sum_{i=1}^{m} f(x_{i}) ...
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22 views

Extremes points of the set of certain increasing convex functions

I am wondering if anyone could help me figure out the following problem. Let $F$ denote the set of all functions $f: [0,1] \to [0,1]$ such that $f$ is weakly convex, increasing, continuous (at end ...
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19 views

The eigenfunctions of $T_{\phi}f:=f\circ\phi$ span $C(X)$ if and only if $\{\phi^{n}:n\in\mathbb{Z}\}\subset C(X,X)$ is equicontinuous.

Let $X$ be a (not necessarily metrizable) compact Hausdorff space. We endow $C(X,X)$ with the topology of uniform convergence. Let $\phi\colon X\to X$ be a homeomorphism and consider the isometric ...
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51 views

Completeness of Metric Space

Consider the set $C^k(\mathbb{R^n},\mathbb{R^m})$ in the metric $\rho$ where $\rho$ is defined by $$\rho(f,g)=\sup\{d(f(x),g(x)),d(f^1(x),g^1(x)),\dots ,d(f^k(x),g^k(x))|x\in \mathbb{R^n}\}.$$ How can ...
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1answer
29 views

For a space X and metric space Y, if X is compact then uniform and compact convergence topology coincides [closed]

In the book Topology by Munkres, there is a theorem: Let $X$ be a space and $(Y,d)$ be a metric space. For the function space $Y^X$ ,one has the following inclusions of topologies: $uniform\supset ...
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0answers
23 views

Notation for the function space of continuous functions defined on a close interval

I have a quick question about the proper way of denoting the function space of continuous functions defined on a close interval. for example the function space of Legendre Polynomial. I think it ...
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1answer
38 views

Convergence of a sequence in $C^{\infty}_c(U)$ under limit topology

For an open $U \subseteq \mathbb{R}^n$, we give $C^{\infty}_c(U)$ (smooth compactly supported complex-valued functions on $U$) the limit topology from the inclusions $C^{\infty}_K(U)$ over all compact ...
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1answer
63 views

"Asymptotic" functionals on $C^k(\mathbb{R})$

Let $C^k(\mathbb{R})$ denote the vector space of $k$-times continuously differentiable functions $\mathbb{R}\to\mathbb{R}$ (with $k\in\mathbb{N}\cup\{0,\infty\}$), and $C^k_c(\mathbb{R})\subset C^k(\...
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1answer
56 views

Completeness and separability of weighted spaces of continuous functions

Let $w:\mathbb{R}^n\rightarrow (0,1]$ be continuous, with $w(0)=1$ and $w(x)\rightarrow 0$ as $\|x\|\rightarrow \infty$. Consider the vector space containing all continuous functions $f:\mathbb{R}^n\...
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0answers
29 views

Topology on space of differentiable functions on euclidean space

What are the different topologies we can give on the space of all differentiable functions on Euclidean space. And also what topologies can be given to the spaces $C^{k}(U,\mathbb{R}^{m})$ and $C^{\...
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1answer
82 views

Question about the algebraic structure of function space

So, on the linear algebra class it is mentioned that function space is an example of the vector space. By the definition of the vector space, any vector space $(V,+,\cdot)$ both an abelian group and a ...
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34 views

How to prove the induced map; $g_* : C(Y) \rightarrow C(X)$, induced from the map $g \in C(X, Y)$, endowed with fine topology is an embedding.

Considering $X$ as a tychonoff space and $Y$ any metric space, and taking the fine topology on function space $C(Y, \Bbb R)=C(Y)$ and $C(X, \Bbb R)=C(X)$. Then the continuous map $g \in C(X, Y)$ ...
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1answer
20 views

Can we define a reverse mapping for the $\phi : Y \rightarrow C(X, Y)$, as $\phi(y) = y' $, where $y' $ is a constant map in $C(X, Y)$.

For a continuous map $\phi : Y \rightarrow C(X, Y)$, as $\phi(y) = y' $, where $y' $ is a constant map in $C(X, Y)$. Can we define a reverse mapping as $\phi' : C(X, Y) \rightarrow Y$ for a metric ...
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1answer
41 views

How can one the cardinality of function space $C(X, Y)$ equipped with uniform topology, where $X$ is a Tychonoff space and $(Y, d)$ a metric space.

How can one the cardinality of function space $C(X, Y)$ equipped with uniform topology, where $X$ is a Tychonoff space and $(Y, d)$ a metric space. I mean how does the cardinality of $X$ and $(Y, d)$ ...
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1answer
38 views

How to prove that a tychonoff space $C(X, Y)$ equipped with fine topology can be embedded in a tychonoff cube.

As we know that a topological space is Tychonoff if and only if it can be embedded in a Tychonoff cube. We have that a function space $C(X, Y)$ equipped with fine topology where $X$ is a Tychonoff ...
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1answer
44 views

Graph structure on graph homomorphisms

Let $G$ and $H$ be directed graphs. I want to form the following new graph $N$: The vertices of $N$ are the graph homomorphisms from $G$ to $H$; Given $f,g:G\to H$, there is an edge from $f$ to $g$ ...
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0answers
59 views

What are some examples of algebras that satisfy the hypothesis of Stone's theorem?

I have just learned Stone's theorem (at least as is written in baby Rudin): Let $\mathscr{A}$ be an algebra of real continuous functions on a compact set $K$. If $\mathscr{A}$ separates points on $K$ ...
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2answers
56 views

Showing that the derivative function is not continuous w.r.t. the sup norm [duplicate]

Let $\mathcal{C}([-1, 1]) = \{f: [-1, 1] \to \mathbb{R}\mid f \text{ is continuous}\}$, $\mathcal{C}^{1}([-1, 1]) = \{f: [-1, 1] \to \mathbb{R}\mid f' \text{ is continuous}\}$, where both sets are ...
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1answer
32 views

How can we prove for a tychonoff space $X$ and a metric space $(Y, d)$, that a map $\phi : Y \rightarrow C(X, Y)$, is an embedding.

How can we prove for a tychonoff space $X$ and a metric space $(Y, d)$, that a map $\phi : Y \rightarrow C(X, Y)$, $\phi(y) = f$, $f$ is a constant map in $C(X, Y)$ is an embedding. Where $C(X, Y)$ is ...
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1answer
43 views

Non-Metrizability of Compact-Open Topology

Let $X$ be a compact metric space and let $Y$ be a non-metrizable topological space. How can we show that $C(X,Y)$ with compact-open topology is non-metrizable? I was thinking of embedding $Y$ into $...
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0answers
35 views

Where can I find literature regarding cardinal invariants of a function space C(X,Y) endowed with the Uniform or Fine topology?

I am working on Function Spaces as a topological space. I want to get a sample paper which studies the cardinal invariants on the function space C(X,Y) rather than on C(X).
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0answers
25 views

Arbitrarily dilated Triebel-Lizorkin space have equivalence with original Triebel-Lizorkin space?

The original Triebel-Lizorkin space is defined as follows: For $\alpha\in\mathbb{R}$, $0<p<\infty$, $0<q<\infty$ and $\phi\in \mathcal{S}(\mathbb{R})$ whose Fourier transform is supported ...
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1answer
73 views

Function space, Harmonic analysis

I have been just reading "An introduction to harmonic analysis" by Yitzhak Katznelson, and try to finish the problem in it. There is a problem in Chapter 1: Let $B$ be a Banach space on $\...
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1answer
124 views

$\Phi: X \to \mathscr F (J,Y)$ is continuous $\iff$ $\forall j\in J$ we have that $X\to Y$ given by $x \mapsto (\Phi(x))(j)$ is continuous.

Let $(X, \mathscr T_X)$ and $(Y,\mathscr T_Y)$ be topological spaces. GOAL: I wish to prove, Let $J$ be a set and denote by $\mathscr F (J,Y)$ the set of all functions from $J$ to $Y$. We equip it ...
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1answer
80 views

Interprete Banach spaces as subspace of the space of continuous functions on a compact space

At the end of the section of the Banach-Alaoglu theorem my professor wrote in his script: "The B-A theorem can be used to prove that for every Banach space $X$ there exists a compact space $K$ ...
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0answers
85 views

What does the norm tell me in this function space?

Consider the space of real functions $$ X:=\bigg\{e^{\frac{\log^2(s)}{\log(x)}}:x\in(0,1),s\in(0,1]\bigg\}. $$ The map $\rho:e^{\frac{\log^2(s)}{\log(x)}}\mapsto s$ is a way to associate a magnitude ...
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0answers
26 views

Natural topology on a particular subset of the set of continuous maps from $[0,1]^n$ to $\mathbb{R}$

Let $n \in \mathbb{N}^\ast$ and $$F = \{f \in \mathcal{C}([0,1]^n,\mathbb{R}) : f \text{ is convex and } f|_{\mathrm{int}(I)} \text{ is } C^1\}$$ where $\mathcal{C}([0,1]^n,\mathbb{R})$ is the set of ...
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0answers
36 views

Do linear algebra and ring theory give any worthwhile results for spaces of continuous/differentiable functions?

Typically, at the beginning of linear algebra texts (and often ring-theoretic ones as well), by way of example of structures being discussed, one is told that the set of all continuous (or ...
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1answer
37 views

Inexistence of a finite $\frac{1}{2}$-net

Let $C\left( [0;1] \right)$ be the set of all continuous functions $f \colon [0;1] \to \mathbb{R}$, and $\| f \|_\infty := \sup\limits_{0 \leq x \leq 1} |f(x)| $ for all $f \in C\left( [0;1] \right)$....
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1answer
189 views

Standard Basis of Function Space $\mathbb{R}^\mathbb{R}$

I have seen $\mathbb{R}^\mathbb{R}$, the set of functions from $\mathbb{R}$ to $\mathbb{R}$, described as a vector space (with the usual operations $(f+g)(x)≔f(x)+g(x)$ and $(a\cdot f)(x)≔a\cdot f(x)$)...
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0answers
45 views

Problem with orthogonalizing the Laguerre polynomials

Alright, so I ran into a little problem while applying the Gram-Schmidt orthogonalization process. To the functions $\{1,x,x^2,x^3...\}$ over $x\in(0,\infty)$ with weight function $\sigma (x)=e^{-x}$. ...
1
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1answer
214 views

Show that, in $C[0,1]$, the functions with $f(\Bbb Q) \subseteq \Bbb Q$ are dense. [duplicate]

For $C[0,1]$ the space of all continuous functions $f:[0,1]\to\mathbb{R}$, Show that, in $C[0,1]$, the functions with $f(\mathbb{Q})\subseteq \mathbb{Q}$ are dense. I'm having trouble understanding ...
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1answer
51 views

Closeness of a family of functions

I am studying chapter 4 of Geometric Measure Theory book by H. Federer and I have some questions from the following part: Assuming that $X, Y$ are Banach spaces with $\operatorname{dim} X<\infty$ ...
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0answers
43 views

The space of all continuous real functions with compact support

Let $\mathbb{X}$ be a Locally Compact Hausdorff Topological Space. Let $C_c(\mathbb{X})$ be the space of all continuous real functions with compact support. I am looking for a good reference for this ...
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0answers
61 views

The metric on the space of continuous functions

Suppose that $G$ is an open set in $\mathbb{C}$ and $(\Omega,d)$ is a complete metric space then designate by $C(G,\Omega)$ the set of all continuous functions from $G$ to $\Omega$. The natural ...
5
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1answer
153 views

Dimension of $r$-jets of maps from manifolds $M$ to $N$

Differential Topology Hirsch Chapter 2 Section 4 Problem 11: Compute the Dimension of $J^r(M, N)$ $J^r(M, N)$ is the set of all $r$-jets from $M$ to $N$. This is an equivalence class $[x, f, U]_r$ of ...
2
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1answer
57 views

Inequality involving the Besov norm

The $L^p$ norm of a function $f$ is given by $$ \|f\|_{L^p(\mathbb{R}^d)}=\left(\int_{\mathbb{R}^d}|f(x)|^p\,dx\right)^{1/p} $$ Let $\phi$ be a non-negative smooth function supported in $\{x\in\mathbb{...
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1answer
48 views

Is the space of continuous functions compactly generated when the space is?

Let $X$ be compactly generated, i.e. a subset $A$ of $X$ is open in $X$ iff $A\cap C$ is open in $C$ for any compact subspace of $X$; and let $(Y,d)$ be a complete metric space. Is it true that the ...