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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).

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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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Does Fermat’s stationary points theorem generalize to multivariable case?

Let $G$ be an open subset of a real-normed space $X$, and $f:X\rightarrow \mathbb{R}$ be a Frechet differentiable function. Assume that $f$ has a local extremum at a point $M\in G$. Then, is $Df(M)=0$? This is true when $X$ is finite dimensional,…
Rubertos
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Norm in the vector space of polynomials

Let $P$ be the vector space of real valued polynomials over $R$. For any polynomial in $P$ set $p(t)=\sum_{k=0}^n a_kt^k $ and $\rVert p \lVert=\sum_{k=0}^n |a_k| $. I am being asked if the following linear maps $l:P \rightarrow R $ and $T: P…
user249018
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spectral properties of commuting self-adjoint operators

In this question $F$ stands for a complex Hilbert space. Let ${\bf S} = (S_1,...,S_d) \in \mathcal{B}(F)^d$. We recall that $\|{\bf S}\|$ is defined by \begin{eqnarray*} \|{\bf…
Schüler
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Canonical embedding of $X$ in $X^{\prime\prime}$ and functionals of $X^{\prime}$

I have been finding it not so easy to understand the topology in $^{\prime}$ ie dual of $X$ . The most obvious on is the topology defined by operator norm topology . But i am not able to get good intuition about defining weak topology and weak star…
Theorem
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Set of all HB extension is convex.

Let $C$ be the set of all Hahn Banach Extensions of a function. I have shown it to be non empty and closed. But how to show that $C$ is a convex set?
User8976
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Is operator B closed?

Hi I have the following problem: Let $X=C([0,1])$ with the $||\cdot||_\infty$-norm. $$A:D(A)\subset X\rightarrow X, Au:=u' , D(A)=C^1([0,1])$$ $$B:D(B)\subset X\rightarrow X, Bu:=u' , D(B)=C^2([0,1])$$ i) Show: $\overline{D(A)}=X$ ii) Is A…
Tobi92sr
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Equivalent Norms in $C^1 [0,1]$

For $x\in C^{1}[0,1]$ let: $${\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert x \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_1=\lvert x(0) \rvert + \lVert x'\rVert_{\infty}$$…
user249018
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Norm of an operator and its matrix

Let $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be linear map, and define $$\| T\|_1= \sup \{ |T(x)|:x\in \mathbb{R}^n, |x|=1\}.$$ Let $(e_1,e_2,\cdots, e_n)$ be the standard basis of $\mathbb{R}^n$, and $[T]=[t_{ij}]$ denote the matrix of $T$ w.r.t.…
Beginner
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Proof of an inequality regarding Hahn-Banach

Let X be a normed $\mathbb{R}$-space, $\gamma>0$,$(x_i)_{i\in \mathbb{N}}$ a sequence in X and $(a_i)_{i\in \mathbb{N}}$ a sequence in $\mathbb{R}$. Show the equality of the following two statements: i) There is a $F\in X'$ with $||F||_{X'}\le…
Tobi92sr
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Is $K:=\{u\in X|\int_0^1|u(t)|^2dt<1\}$ bounded?

Let $X:=C([0,1];\mathbb{R})$ with the norm $||u||_\infty:=sup_{t\in[0,1]}|u(t)|$. Also let $K:=\{u\in X|\int_0^1|u(t)|^2dt<1\}$ Show that K is convex, symmetric, $0\in K$. And show if K is bounded. I already showed the convexity and that K is…
Tobi92sr
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Power series of holomorphic maps into Banach spaces

In my functional analysis class we defined a map $x:\Omega \to F,$ where $\Omega\subset \mathbb{C}$ is open and $F$ is a complex Banach space, to be differentiable in $z_0\in \Omega$ if the limit \begin{equation}\lim_{z\to z_0}\frac{1}{z-z_0}\left(…
Frieder Jäckel
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The annihilator of a normed vector space

Given a normed vector space $X$, consider a subspace $Z$ of its continuous dual $X^{\ast}$. I want to show that if the annihilator of $Z$ is zero, then $Z$ is $weak^{\ast}ly$ dense in $X^{\ast}$. How should I prove this? I'd like to use Hahn Banach,…
Keith
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How can I solve this Equation?

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $K:\Omega\times\Omega\rightarrow\mathbb{R}$ with $\|K\|_{L^2(\Omega\times\Omega)}<1.$ How can I show that the following equation has only the zero solution? $$u(x)=\int_\Omega…
Tomás
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Prove that Space of Lipschitz continous functions is Banach space

Let $X$ be the vector space of Lipschitz continous functions, $[0, 1] \rightarrow \mathbb{R}$. For $x\in X$ set $$\Vert x \Vert_{Lip}=\vert x(0)\vert+sup_{s\neq t}\left\vert \frac{x(s)-x(t)}{s-t}\right\vert.$$ I need to prove: $\Vert x…
user249018
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Show that the Newton-Raphson iteration is a contraction under certain conditions

The Newton–Raphson method for finding roots of a differentiable function $f : \mathbb{R} \to \mathbb{R}$ is to iterate the function $$x \mapsto g(x):=x - \frac{f(x)}{f'(x)}$$ Where the following conditions is assumed on $f$ $f$ has a continous…
N3buchadnezzar
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