# 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).

31,913 questions
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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 ...
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### Why do we need projection in the definition of the Stokes operator?

$\DeclareMathOperator{\div}{div}$ $\def\bu{\mathbf{u}}$ Let $D$ be the square $[0,1]^2$ and consider the following space: $$V:=\{\bu: \bu\in H^2(D)^2, \div \bu=0, u|_{\partial D}=0 \}.$$ Introduce ...
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### Is it possible for a sequence of matrices to have pointwise but not uniform convergence?

Is it possible for a sequence of matrices to have pointwise but no uniform convergence? The norm for the matrices is the operator norm.
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### Show that Lipchitz functions space is Banach

Let X be a Banach space. Show that $L = \{ f:X \to \mathbb{R}: f \text{ is lipschitz and } f(0)=0 \}$ with norm: $$||f||_{lip}= sup \left\{ \frac{|f(x)-f(y)|}{||x-y||}; x \neq y \in X \right\}$$ ...
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### Determining if $f \in BV([0,1])$.

I am reading about Riemann-Stieltjes Integration in Carother's Real Analysis. We find that functions of Bounded Variation provide us with a rich class of integrators. Therefore, I am trying to learn ...
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### Is the integral equal to zero?

Consider $$\int_{S_r} \left(\frac{\partial\phi}{\partial\eta}\right)^2 (x\cdot\eta)\ d\sigma$$ Where $S_r$ is the sphere of radius $r$ and centered at zero, $\phi\in L^2(\mathbb{R}^n)$ and $\eta$ is ...
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### Sufficiency in the proof that $L^p(\mu)$ is complete

In the proof that $L^p(\mu)$ is complete for $p\in[1,\infty]$ (as done in Saxe, Theorem 3.21 or in Folland, Theorem 6.6, the latter of which is outlined here) we make use of the following completeness ...
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### Does the series corresponding to a Cauchy sequence **always** converge absolutely?

Let $X$ be a normed vector space and consider a Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ in $X$. Is it true that the corresponding series of our Cauchy sequence, $\sum_{i=1}^\infty x_i$, always ...
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### Hilbert Transform: limit of xHf(x)

In Terence Tao's notes page 1, cited below, he mentions that it is easy to see that $\lim_{|x| \to \infty} xHf(x) = \frac{1}{\pi}\int f$ where $f$ is a Schwartz function and $H$ is the Hilbert ...
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### What is the defining properties of the heat kernel?

According to https://en.wikipedia.org/wiki/Fundamental_solution, the fundamental solution $u$ is defined as the solution of $Pu = \delta$. In the meantime, we know that the fundamental solution of the ...
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### How are vector norms and function norms related? [closed]

I'm a little bit confused, so can somebody explain how are vector norms and function norms related?