# 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).

32,040 questions
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### Conditional convergence of series $\sum_{n=1}^{\infty}a_n(\frac{k}{n}-k^{n+1}\frac{\zeta(n+1)}{n+1}),0<k\le 1$

Does the series $\displaystyle \sum_{n=1}^{\infty} a_n\left(\frac{k}{n}-k^{n+1}\frac{\zeta(n+1)}{n+1}\right),0<k\le 1$ converges conditionally given that $\sum_{n=1}^{\infty}|a_n|<\infty$. ...
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### $a=cc^*c$ for some $c$. $a \in A$ a $C^*$ algebra.

Let $A$ be a $C^*$ algebra. Let $a \in A$, then there exists $c \in A$ such that $a=cc^*c$. This fact is used from example (1) of Prop 4.25. How does one show this?
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### Want to show that sequences in little l 1 can be represented by orthonormal basis

I want to show that for a sequence $a=\{a_i\}^{\infty}_1 \in l^1$ I can write $$a= \sum ^\infty _{i=1} a_i e_i$$ For $\{e_i\}^\infty_1 \in l^1$ where the ith term is 1 and all else zero. I know for ...
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### Functional differentiation with Grassmann variables

I want to calculate: $$\frac{\partial}{\partial (x^\rho+i\eta\, \psi^\rho)}[f_\mu(x)+i\eta\, \psi^\nu\partial_\nu f_\mu(x)]$$ where $x^\mu(\tau)$, $\psi^\mu(\tau)$ are commuting and anti-commuting ...
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### Hessian of inner product

Let $f:\mathbb R^n \rightarrow \mathbb R^n$ be a $C^2$ functions and consider the function $h:\mathbb R^n \rightarrow \mathbb R$ given by $$h(x):=\langle f(x),f(x) \rangle.$$ I am wondering whether ...
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### If $\mathcal D(\overline A)$ is dense, are we able to conclude that $\mathcal D(A)$ is dense?

Let $(\mathcal D(A),A)$ be a closable linear operator on a $\mathbb R$-Banach space $E$ and $(\mathcal D(\overline A),\overline A)$ denote its closure. If $\mathcal D(\overline A)$ is dense, are we ...
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### Definition of closed

In topology we say that a set is $A$ is closed if its complement $A^\complement$ is open. Now if we have a set $A$ and an operation $*$ we say that $A$ is closed under this operation if $x,y \in A$ ...
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### $H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$?

In a paper I see that the authors used $H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$ where $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary. I think that this imbedding holds ...
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### How are the two versions of the Lumer-Phillips theorem given in the books of Engel/Nagel and Pazy related?

Let $E$ be a $\mathbb R$-Banach space and $(\mathcal D(A),A)$ be a densely-defined dissipative linear operator on $E$. In the book of Engel and Nagel, I've found the following verison of the Lumer-...
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### Continuity of Linear Operator Between Hilbert Spaces

Note: Please do not give a solution; I am curious to understand why my solution is incorrect, and would prefer guidance to help me complete the question myself. Thank you. Let $\mathcal{H}$ be a ...
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### Set $X$ such that $2^c \leq |X|\leq |(\ell^\infty)^*|$ [duplicate]

I want to prove that $2^c\leq|(\ell^\infty)^*|$. My question is: Is there a set $X$ with $2^c\leq|X|$ such that $f:X\to(\ell^\infty)^*$ is an injective function or $g:(\ell^\infty)^*\to X$ is a ...
Let $V,W$ be complex inner product spaces. Suppose $T: V \to W$ is a linear map, then we define $$\|T\|:=\sup\{\|Tv\|_{W}:\|v\|_{V}=1\}$$ where $\|v\_{V}\|:=\sqrt{\langle v,v\rangle}$ and $\|Tv\|_{W}... 1answer 38 views ### Show that there is a solution of the Laplace equation$(\mu-A)p=f$Let$C_0(\mathbb R)$denote the space of continuous functions vanishing at infinity equipped with the supremum norm,$B$be a contractive linear operator on$C_0(\mathbb R)$and $$Af:=\lambda(Bf-f)\;\;... 1answer 49 views ### Finite rank operators on Hilbert spaces Let H be a Hilbert space. Question 1: Are all rank one operators from H to H is of the form$$T:H\rightarrow H, x \mapsto \langle x,u\rangle v$$For some$u,v \in H$. Question 2:... 1answer 34 views ### Rudin's functional analysis, theorem 4.23 (existence of a certain sequence) If$X$is a Banach space,$T \in \mathcal{B}(X)$,$T$is compact, and$\lambda \neq 0$, then$T - \lambda I$has closed range. Proof with questions below Proof: By (d) of Theorem 4.18,$\text{dim }...
Let $u^+$ denote the positive part of the function $u$ on a bounded domain $\Omega.$ If $u \in C^{0,\alpha}(\bar \Omega)$, is also $u^+ \in C^{0,\alpha}(\bar \Omega)$?