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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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weak convergence in $\ell^{\infty}$

Let $e_n = (0,0,\dots,1,0,0,\dots)$,(i.e. the n-th component is 1). Show that $e_n\rightharpoonup 0$ in $\sigma(\ell^{\infty},(\ell^{\infty})')$. I'm having trouble because the dual of $\ell^{\infty}$ is not $\ell^1$. Could someone give me some…
QD666
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Norm of $T:(C[0,1],||.||_{\infty})\to \mathbb R,$ $T(f)=\int_0^1tf(t)$

Let $C[0,1]$ denote the set of all continious real valued function on the interval $[0,1]$ and $T:(C[0,1],||.||_{\infty})\to \mathbb R$ be defined by $T(f)=\int_0^1tf(t)$ for all $f \in C[0,1]$. Then find $||T||$? My attempt: i take …
jasmine
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Show that there exists $r>0$ such that $\{y\in Y:||y||\leq r\}\subset \bar{L}.$ (Open Mapping Theorem).

Suppose $X$ and $Y$ are Banach Spaces and $T$ is a surjective bounded linear operator in $B(X,Y).$ Let $$L = \{T(x):x\in X \text{ and }\|x\|\leq 1\}$$ with closure $\bar{L}.$ Then show that there exists $r>0$ such that $\{y\in Y:\|y\|\leq…
Student
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A bounded linear operator that is close to a unitary operator should be invertible.

Suppose that $T,U$ are bounded operators on a Hilbert space $H$, with $U$ unitary. If $\|T-U\| < 1$, then show that $T$ is invertible. Injectivity is easy: Suppose $z \in \ker T$, then $\| (T-U)z \| < \|z\|$ for $z \neq 0$ which implies that $z =…
z.z
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The image of a multiplication operator has to be either the whole space or of first Baire Category

Let $\phi \in C[0,1]$ and $T_{\phi}: C[0,1] \to C[0,1]$ to be the multiplication operator such that $T_{\phi}(f) = \phi f$, then either the range of it is the whole space or it is of the first Baire catergory. Well, by Baire category theorem, if…
z.z
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$I[G(A')^{\bot}]=G(A)$ - Proof of Theorem 3.24, Brezis " Functional Analysis,Sobolev spaces and partial differential equation"

The statement of the theorem is: Let $E$ and $F$ be two reflexive Banach spaces. Let $ A:D(A) \subset E \rightarrow F$ be an unbounded linear operator that is densely defined and closed. Then $D(A')$ is dense in $F'$. Thus $A''$ is well defined (…
F. Buseghin
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separability of the Schatten-von-Neumann class

My interest is to know whether the assertion : the class of Schatten-von Neumann of operators over a separable Hilbert space is separable with respect to the $\|\cdot\|_p$ norm. is correct?? Thanks!
Khadi Gan
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Open Mapping Theorem, Brezis, step 2

I'm trying to understand the second step of Open Mapping Theorem's proof by Brezis ( Functional Analysis, Sobolev Spaces and Partial Differential Equations ). The statement of the theorem is the following. Let $E$ and $F$ be two Banach spaces and…
F. Buseghin
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Proof of Hahn Banach theorem geometric form of Brezis : if $A$ and $B$ is disjoint, why introducing $C=A-B$?

Let $A\subset E$ and $B\subset E$ two convexe sets, non empty and disjoint. Suppose $A$ open. There is a closed hyperplan that separate $A$ and $B$. In the book "Analyse fonctionnelle : théorie et application" (sorry it's in french), page 6…
Peter
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Calculate the operator norm defined on L2

For $f \in L^2([0, 1]) $ and $x \in [0, 1]$ we have the following $$ A(f)(x) = \int_0^1 (x - y) f(y)dy $$ In this particular exercise I have to calculate the operator norm $\| A \|$. We know that from a theorem that if $A$ is compact and…
Supersalt
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In what Hahn Banach theorem is so important?

Hahn—Banach theorem says that : Let $E$ a $\mathbb R-$vector space and $p:E\to \mathbb R$ a sub-linear application. Let $G$ a subspace of $E$ and $g:G\to \mathbb R$ a linear application. Then there is $f:E\to \mathbb R$ s.t. $f(x)=g(x)$ for all…
user380364
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Let $E$ a vector space and $f\in E^*$. Why $\|f\|:=\sup_{x\in E, \|x\|\leq 1}|f(x)|=\sup_{x\in E, \|x\|\leq 1}f(x)$?

Let $E$ a vector space and $f\in E^*$. By definition $$\|f\|_{E^*}:=\sup_{x\in E, \|x\|\leq 1}|f(x)|.$$ But why do we have $$\|f\|_{E^*}=\sup_{x\in E, \|x\|\leq 1}f(x) \ \ ?$$ Because linear functional are not positive, do they ?
Peter
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Proving compactness of $A:l_2\to l_2\:\:Ax=(0,x_1,\frac{1}{2}x_2,...,\frac{1}{n}x_n,...)$

Problem: Show that the operator $A:l_2\to l_2\:\:Ax=(0,x_1,\frac{1}{2}x_2,...,\frac{1}{n}x_n,...)$ is compact but has no pontual spectrum. My solution: $(0,x_1,\frac{1}{2}x_2,...,\frac{1}{n}x_n,...)=(\lambda x_1,\lambda x_2,...,\lambda…
Pedro Gomes
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Weak operator topologies on $B(X')$

Let $X$ be a Banach space with dual $X'$. On the space of bounded operators $B(X')$ on $X'$ we can define the following two weak operator topologies defined by the seminorms: $T \mapsto |\langle Tx', x'' \rangle| = |x''(Tx')|$ for $x' \in X$, $x''…
yada
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Continuous linear transformation on normed linear spaces with $T(x)=y$

Let $X,Y$ be normed linear spaces and let $x(\neq0)\in X,y(\neq0)\in Y$. prove that there exists $T\in \mathscr{B}(X,Y)$ such that $Tx=y$. I thought of a constant map but that will be not linear and continuous. Please give me idea how to construct…
ravi yadav
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