# Tagged Questions

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, ...

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### Set of diffeomorphisms on a manifold

It is well known that given a compact smooth boundaryless manifold $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r \geq 1$, is open in $C^{r}(M)$, the set of continuous functions (for ...
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### Find norm of operator

I have a linear functional $$A: L_2[0,2] \to \mathbb R, Ax = \int_0^2(t^2+2)x(t)dt$$ I need to find $C$, trying to measure $C$ and $||Ax||$ to find it, but how can I do it in this problem?
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### Extending a functional with same norm

I have: $X = <\mathbb R^2, ||(x,y)|| = \sqrt{4x^2+y^2}>, L = \{(2x,3x), x \in\mathbb R\},$ $\phi_0 \in L^* : \phi_0 (2x,3x) = -2x$ I need to extend $\phi_0$ on $X$ without changing norm (...
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### Density of bounded functions in $L^1(0,T;L^1(M))$?

Let $u \in L^1(0,T;L^1(M))$ where $M$ is a compact Riemannian manifold. Is it possible to find $u_n$ such that $u_n \to u$ in $L^1(0,T;L^1(M))$ and $u_n$ are bounded everywhere or almost everywhere on ...
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### Find a norm of operator form l1 to l1

I have an operator $A: \ell_1 \to \ell_1, Ax = (x_1+x_2, x_1-x_2, x_3,...,x_k,...)$ AFAIK, norm of $\ell_1$ is $\sum_{n=1}^{\infty}|x_n|$ How to find a norm of this operator?
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### Gaps in my proof of the Arzela-Ascoli Theorem - help and expertise greatly appreciated for an alternate formulation.

I have a general outline of the proof of the Arzela-Ascoli Theorem but have trouble filling in the gaps of the theorem. I have posted the entire general method I believe to be correct below. I was ...
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### Find the norm of the operator $A:L_2[0,2] \rightarrow L_2[0,2]$ defined by $(Ax)(t) = t \, \operatorname{sgn}(t-1) x(t)$

I have operator: $\boldsymbol{L}_2[0,2] \to \boldsymbol{L}_2[0,2], ( Ax)( t ) = \boldsymbol{t} \operatorname{sgn}(t-1)x(t)$ I need to find operator norm or say that operator isn't bounded. ...
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### A topological vector space with countable local base is metrizable

I feel confused by the proof of the following theorem in Rudin 2/e: Theorem 1.24 If $X$ is a topological vector space (t.v.s.) with a countable local base, then there is a metric $d$ on $X$ s.t. ...
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### Nontrivial functionals on $l^\infty$ vanishing on $c_0$

I understood that the dual of $c_0$ is a proper subspace of the dual of $l^\infty$, by Hahn-Banach theorem. But how can I find functionals in $(l^\infty)^*$ vanishing on $c_0$?
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### Maximal norm of tensor product

Definition 3.3.3. (Maximal Norm) Given $A$ and B (two C*-algebra), we define the maximal C*-norm on $A \odot B$ to be $$||x||_{max}=sup\{||\pi(x)||: \pi: A\odot B\rightarrow B(H) ~a~*-homomorphism\}$$...
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Most of the questions are more measure theory and integration related but I need to give some context, so I will now. Consider the quasilinear 2nd-order partial differential equation $$-\text{div}(a(... 0answers 81 views ### Does absolute continuity of measures imply a relation between the L_p spaces? Say (X,\mathcal{B},\mu) is some measure space, and let \sigma be some other measure on (X,\mathcal{B}) such that \sigma\ll\mu. What can one say about the relation between L_p(\mu) and L_p(\... 1answer 43 views ### Inequality ||f-g|| < \epsilon \Rightarrow |E[f] - E[g]| < \epsilon Let C(X) be the space of continuous bounded functions on some metric space (X,d). Can it be shown that if ||f-g||_\infty < \epsilon if follows that | \int f \, \text{d}P - \int g \, \text{d}... 1answer 145 views ### Uncountable disjoint union of measure spaces Let (a,b) be an interval. Let (A_i, \Sigma_i, \mu_i) be a measure space for each i \in (a,b). Is it possible to put a measure space on the disjoint union$$\bigcup_{i \in (a,b)}\{i\}\times A_i?$... 1answer 23 views ### What is the definition of$\min$,$\max$of functions,$f_i$? I have a couple of questions: What is the definition of the expressions on the right-hand side? Each$f_i : X \to \overline{\mathbb{R}_+}$$$h = \max(f_1, f_2, f_3,...f_n)$$ $$h = \min(f_1, f_2, ... 1answer 83 views ### Question about mollifiers. So here is my problem, Let \rho \in C^\infty (\mathbb{R}^n,R) with \rho\geq 0, \rho(x)= 0 \; \forall \|x\|\geq 1 and \int_{\mathbb{R}^n}\rho(x)dx=1. Further, consider the linear map K_f:L^p\... 1answer 64 views ### how does semi-inner product and symmetric positive semi-definite bilinear form are different? Given vector space V over scalar field \mathbb R, I wonder if two definitions "semi-inner product" and "symmetric positive semi-definite bilinear form" are actually equivalent. The definition of "... 1answer 182 views ### Euler's Refutation of Fermat's Conjecture Fermat postulated that all numbers of the form$$2^{2^n}+1$$are prime (where n = any integer). Then Euler came along with a rather ingenious proof that this was not, in fact the case. I came across ... 0answers 39 views ### Orthonormal set problem A)For the First three member of (x_0, x_1, x_2, ... ) with respect to$$x_{j}(t)=t^{j}$$in [-1,1] , use the inner product function below to make them orthonormal.$$\langle x,y\rangle =\int_{-1}^... 1answer 37 views ### How to verify$H\otimes K \cong \bigoplus\limits_{i\in I}H$Let$H,~K$be the Hilbert space. if$\{v_{j}\}_{j\in J}\subset H$and$\{w_{i}\}_{i\in I}\subset K$are the orthonormal bases, then how to construct the isomorphic mapping:$H\otimes K \rightarrow \...
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I have already tried but I failed. I can't show it is. I used this way: $| (Kf)_n |^2 \leq c_n \|f\|^2$, and therefore $\|Kf\|^2 \leq \|c\| \|f\|^2$, so that $\|K\| \leq \sqrt{\|c\|}$ is bounded, ...
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### Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
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### Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
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### Necessity of hypothesis in distance from a set in an inner product space

In Kreyzig's Functional Analysis book, one theorem in inner product spaces is about the existence and uniqueness of a minimal point from a set. 3.3-1 Theorem (Minimizing vector). Let $X$ be an ...
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### Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
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### Is the linear operator $T_2^{-1}T_1 :U_1 \to U_2$ bounded if $T_1\in L(U_1,H)\ \ T_2 \in L(U_2,H)$ $\mathrm{ker}\ T_2=\{0\}$?

Let $U_1, U_2, H$ are Hilbert spaces. $T_1\in L(U_1,H)\ \ T_2 \in L(U_2,H)$, $\mathrm{ker}\ T_2=\{0\}$, and the image of $\ T_1,T_2$ are the same, i.e. $\mathrm{Im}\ T_1=\mathrm{Im}\ T_2$, My ...
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Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega)$ and consider $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some boundary conditions. Consider the weak form $$\int_{\Omega}u_t\... 3answers 1k views ### The difference between hermitian, symmetric and self adjoint operators. I am struggling with the concept of hermitian operators, symmetric operators and self adjoint operators. All of the relevant material seems quite self contradictory, and the only notes I have to go ... 2answers 600 views ### Reinventing The Wheel - Part 1: The Riemann Integral [closed] Preface The core of any notion of integral is some sort of weighted sum:$$\sum b\mu(A)$$Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones:$$\{...
Let $K$ and $L$ be two compact set and $T$ is an linear onto isometric from $C(K)$ to $C(L)$. My question is that $T(1)$ is the identity map in $C(L)$, where 1 is the identity map in $C(K)$ . give me ...