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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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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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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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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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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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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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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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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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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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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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 ...
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 $\... 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 ... 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$K$... 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 ... 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 ... 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 ... 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)$.... 1answer 189 views Standard Basis of Function Space$\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)$)... 0answers 45 views Problem with orthogonalizing the Laguerre polynomials Alright, so I ran into a little problem while applying the Gram-Schmidt orthogonalization process. To the functions$\{1,x,x^2,x^3...\}$over$x\in(0,\infty)$with weight function$\sigma (x)=e^{-x}$. ... 1answer 214 views Show that, in$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}$are dense. I'm having trouble understanding ... 1answer 51 views Closeness of a family of functions I am studying chapter 4 of Geometric Measure Theory book by H. Federer and I have some questions from the following part: Assuming that$X, Y$are Banach spaces with$\operatorname{dim} X<\infty$... 0answers 43 views The space of all continuous real functions with compact support Let$\mathbb{X}$be a Locally Compact Hausdorff Topological Space. Let$C_c(\mathbb{X})$be the space of all continuous real functions with compact support. I am looking for a good reference for this ... 0answers 61 views The metric on the space of continuous functions Suppose that$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$. The natural ... 1answer 153 views Dimension of$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$of ... 1answer 57 views Inequality involving the Besov norm The$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{...
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 ...