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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 .

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Is any completely continuous operator in Hilbert space an operator with an infinite dimensional image?

Let $A$ be an operator in Hilbert space, and let the elements in H be composed of an orthonormal system of eigenvectors $\phi_n$. Then any completely continuous operator satisfies the notion $$\lim_{n\longrightarrow \infty}\lambda_n=0$$. And we…
Luthier415Hz
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Prove the existence of a certain continuous linear functional

I have the following homework problem: Let $E$ be real normed space and consider $x,y\in E$ such that $$\Vert x\Vert=\Vert y\Vert=1$$ $$\Vert 2x+y\Vert=\Vert x-2y\Vert=3$$ Prove that there is $f\in E^{\star}$ such that $\Vert f\Vert=1$ and…
joelittledrop
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How to prove this linear operator is bounded?

Suppose $X$ is a Banach space, $T:X\to X^*$ is a linear operator, which satisfies $$\langle Tx,x\rangle\ge0,\quad\forall x\in X.$$ How to prove $T$ is bounded? The question will be easy if $X$ is a complex Banach space. I'm wondering whether we can…
zcsb
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Prove that $A$ is bounded and find its norm

Consider $A$ is a linear operator: $E \rightarrow E$ on Banach space $ E=C([0,1], \mathbb{R}) $. $(Af)(t) = \int_0^1{(t^2+s^2)f(s)ds}$ Prove that $A$ is bounded and find its norm. My solution: $||Af||=\sup \limits_{t \in [0,1]}|(Af)(t)|=\sup…
Olha
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Is there a normed vector space (n.v.s) that contains all the others n.v.s?

I was thinking about the following question: Can be possible to exist a normed vector space $(E,||\cdot||)$ (let us think all vector spaces are over $\mathbb{R})$ such that if $(F,||\cdot||_F)$ is another normed vector space then there is a linear…
Jacaré
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Direct sum bornology

The direct sum bornology is defined the following way: Let $I \neq \emptyset$ be a index set and $(X_i,\mathcal{B}_i)$,$i \in I$ a family of bornological vector spaces, $X = \bigoplus_{i \in I}X_i$ and $\iota_i : X_i \to X$,$i \in I$ the canonical…
Orb
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Notation for the space of functions with $L^2$ derivative, but may themselves not be $L^2$

Consider the space of functions $$f: \mathbb{R} \to \mathbb{R}$$ such that $$f(0) = 0$$ and $$\int_{-\infty}^{\infty} f'(x)^2 dx < \infty$$. Does this sort of space have a common notation or name? Note that this is not the same space as $L^2_1$,…
graviton
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Finding a good example of an element of a given dual space

Let $$\{X=c_0=\{x=(x_j):\lim_{j\longrightarrow\infty}x_j=0\}$$ and the norm be $$||x||_\infty=\underset{1\le j\le\infty}{max}|x_j|.$$ I want to find two examples of elements of the dual space $X^*$. The first is : $$s_n=\frac{n+1}{(n+2)^2}$$, since…
Luthier415Hz
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continuous map with finite limit condition

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set . Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $\displaystyle\lim_{s\to\pm\infty}\frac{f(s)}{s}=f_{\pm}$ with $f_{\pm}$ finite . Show that the mapping $u\to f(u)$ defined…
am_11235...
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Let $f:\mathbb{R}\to\mathbb{R}$ be continuously differentiable,a sequence of polynomials $p_n$ s.t. $p_n\to f$ and $p_n'\to f'$, uniformly on $[a,b]$

Let $f:\mathbb{R}\to\mathbb{R}$ be continuously differentiable. Show that for every bounded interval $[a,b]$,there exists a sequence of polynomials $p_n$ such that $p_n\to f$ and $p_n'\to f'$, uniformly on $[a,b]$. My solution: I know that by using…
djc
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Subset of a Banach space

If $X$ is the space of all bounded continuous Real valued functions defined on $\mathbb{R}$ . For every $f$ belonging to $X$ define $$ \|f\|_\infty=\{\sup|f(x)| ,x \text{ belongs to } \mathbb{R}\}. $$ I have proved that $X$ is a Banach space. The…
Mohammad El Mahdi Jaafar
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Let $T$ be a map from $\ell^3 \to \ell^1$ and $S$ be a map from $\ell^1 \to \ell ^3$. Are these maps continuous?

Let $T:(x_n)_{n=1}^\infty \mapsto (n^{-1}x_n)_{n=1}^\infty$ be a map from $\ell^3 \to \ell^1$ and $S:(x_n)_{n=1}^\infty \mapsto (\log(n+2)x_n)_{n=1}^\infty$ be a map from $\ell^1 \to \ell ^3$. Are these maps continuous? One needs to show…
Epilogue
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Prove that if $\left\{f_k\right\}_{k=1}^{\infty}$ is a Bessel sequence in $\mathcal{H}$, then $\sum_{k=1}^{\infty} c_k f_k$ converges unconditionally.

I would like to prove the following: Let $\mathcal{H}$ be a Hilbert space. If $\left\{f_k\right\}_{k=1}^{\infty}$ is a Bessel sequence in $\mathcal{H}$, then $\sum_{k=1}^{\infty} c_k f_k$ converges unconditionally for all…
Mark
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Are Fourier transforms over $L^1(\mathbb{R})$ themselves in $L^1(\mathbb{R})$?

I know that $\hat{f}(\omega)$ is in $C_0(\mathbb{R})$ (continuous with limit $0$ as $n \to \infty$), but that doesn't necessarily mean that it is in $L^1(\mathbb{R})$, from my understanding. Is there some other data that shows that it is? Also, with…
Anon
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Finding the spectrum of $(Tf)(t)=\max[0,\cos(t)]f(t)$ for $f\in L^2[-\pi,\pi]$

I've nearly completed finding the spectrum of $(Tf)(t)=\max[0,\cos(t)]f(t)$ for $f\in L^2[-\pi,\pi]$, but I cannot seem to finish it off. Here's what I've done so far: I used the fact that $||T||=1$ and that the spectrum is closed and bounded by…
Anon
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