# Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

31,968 questions
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### n-times commutator of a function with $-\Delta +v$

Let $f$, $v$ be smooth function and let us assume that $v$ is also bounded. I'm needing somehow a formula for the $n$-times commutator of $f$ (interpreted as multiplication operator) with the ...
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### Spectral decomposition of the resolvent map

Let $P_\Omega$ be a projection valued measure and let $R_A(z)=(A-z)^{-1}$ be the resolvent map. It can be shown that $$R_A(z)=-\sum_{j=0}{\frac{A^j}{z^{j+1}}}$$ whenever this series is defined. My ...
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### Michael's selection theorem

Michael's selection theorem states that a lower hemicontinuous multivalued map with nonempty convex closed values $\displaystyle F\colon X\rightrightarrows E$ from a paracompact space $X$ to a Banach ...
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### $X,Y$ Banach, $V \subset X$ a linear subspace, $T:V\to Y$ closed, then $T$ bounded $\iff$ $V$ closed

Let $X$ and $Y$ be Banach spaces and let $V \subset X$ be a linear subspace. Let $T: V \to Y$ be a closed linear operator. Show that $T$ is bounded if and only if $V$ is closed. For the direction $V$ ...
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Please I dont understand this. I have: $\parallel \nabla m_n \parallel_{L^{\infty}(\mathbb{R}^+, L^2(\Omega))}\leq C$ $\parallel \frac{\partial m_n}{\partial z} \parallel_{L^{\infty}(\mathbb{R}^+,... 1answer 41 views ### Uniform inequality for a continuous function Let$f(x,y)\in \mathcal{C}([a,b]\times[c,d])$such that $$\exists \xi\in (a,b) : f(\xi,y)\neq 0, \forall y\in [c,d].$$ By the continuity of$f$, we have $$|f(\xi,\cdot)|\geq \min\limits_{[c,d]} |f(\... 0answers 32 views ### What is the usual definition of the spectral measure for a nonnegative self-adjoint operator on a Hilbert space? Let H be a \mathbb R-Hilbert space and (H_\lambda)_{\lambda\ge0} be a spectral decomposition of H (see below). Now, let$$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\... 0answers 32 views ### For a compact Riemannian manifold$M$,$L^2(M)$is spanned by the eigenfunctions of the Laplacian. In some paper I read the following statement: For a compact Riemannian manifold$M$and the corresponding Laplace-Beltrami operator$\Delta$on$Mwe have, that L^2(M) = \widehat{\bigoplus_{\... 0answers 18 views ### Shift operator on the double-size Hilbert space \ell^2(\mathbb{N}^*)\oplus \ell^2(\mathbb{N}^*) It is well known that, the right shift operator is given by \begin{align*} A_1\colon \ell^2(\mathbb{N}^*) & \rightarrow \ell^2(\mathbb{N}^*) \\ (x_1,x_2,\cdots)&\mapsto (0,x_1,x_2,\cdots), \... 1answer 15 views ### Show that resolvant is analytic outside the spectrum Let T be a bounded operator on Hilbert space \mathcal{H}. Show that R_{\lambda}=(T-\lambda)^{-1} is an analytic function on open set \rho(T)=\mathbb{C}\setminus sp(T). I know R_{\lambda}=-\... 1answer 35 views ### The tensor product of two blocks of positive operators is positive LetT = \begin{bmatrix} T_{11} & T_{12}\\ T_{21} & T_{22} \end{bmatrix},\quad S = \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix}$$be two positive operators on E\... 1answer 50 views ### Prove \|T\| = \sup_{\|x\| < 1} \|Tx\| Let X, Y be Banach spaces. And T \in B(X\rightarrow Y). Prove that$$\|T\| = \sup_{\|x\| < 1} \|Tx\|$$Discussion Having trouble seeing how to handle some of these ideas below. Please let me ... 0answers 19 views ### Weak Convergence from Strong Convergence Let \Omega \subset \mathbb{R} be a bounded domain, v \in H_{0}^{1}(\Omega) such that ||u_{n}-v||_{H_{0}^{1}(\Omega)}\to 0 as n\to\infty for a bounded sequence \{u_{n}\}_{n\in\mathbb{N}} \... 2answers 40 views ### Deforming a linear map a little preserves surjectivity Let X be a Banach space, and A: X\to X be a surjective linear map. Define$$\eta(A) = \{\lambda\in \mathbb{C}: A - \lambda I \text{ is surjective}\}$where$I: X\to X$is the identity. Show that$...
Consider a finite Borel measure $\mu$ on $\mathbb R$. The Fourier transform $\hat{\mu}$ of $\mu$ is defined by $\hat{\mu} (\xi)= \int _{\mathbb R} e^{-ix\xi} d\mu(x)$. I would like to prove the ...