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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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Existence of non-zero $\sigma$ -finite $R^{(\alpha)}$-invariant Borel measure in $R^{\alpha}$

Let $R^{\alpha}$ be a vector space of all real-valued functions defined on a non-empty parameter set $\alpha$. Let $\cal{B}( R^{\alpha})$ denotes a Borel $\sigma$-algebra of subsets of $R^{\alpha}$ generated by Tychonoff topology. Let a vector…
George
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Finding explicitly the operator norm of $T u = \sin (t) u $

I have the operator $$T: L^2(0,\pi)\to L^2(0,\pi), \ \ \ Tu=\sin(t)u.$$ I can show that the operator norm is bounded by $1$ but I can't find any function to show the norm is exactly that. I am asked to find It explicitly. Can anybody give me some…
lucmobz
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If $X^{*}\simeq Y^{*}$, then $X\simeq Y$?

Let $X, Y$ be Banach spaces s.t. $X^{*}\simeq Y^{*}$. Is it true that $X\simeq Y$? This is clearly true for finite dimensional spaces, but I can't certain that this holds for infinite dimensional case since $X^{**}$ is not isomorphic to $X$ in…
Seewoo Lee
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Exercise 4.13 in Brezis: L1 limit

I am having trouble with the following exercise: Using $||a+b|-|a|-|b|| \leq 2 |b|$, prove that given a sequence $(f_n)_n \subset L^1(\Omega)$ with $f_n(x) \to f(x)$ a.e. $(f_n)$ is bounded in $L^1$, i.e. $\|f_n\|_1 \leq M$ for some $M\in…
Philipp Wacker
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$\|f\|\geqslant \|u\|$ functional on ${L_p}_{[a,b]}$

Consider the following functional $f(x)=\int_\limits{a}^{b}x(t)u(t)dt$, where $x(t)\in {L_p}_{[a,b]}$ and $u(t)\in {L_q}_{[a,b]}$ The space ${L_q}_{[a,b]}$ is the conjugate of ${L_p}_{[a,b]}$ which implies $\frac{1}{p}+\frac{1}{q}=1$. It easy to…
Pedro Gomes
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Hahn-Banach theorem corollaries

Notes :Corollary: Let $L\subset X$ be a Banach space, then there exists a complement of $L$ that is closed and can be defined by linear functionals linearly independent. Proof: Consider $e_1,e_2...e_n$ a basis of $L$. Then $f_i(e_k)=\delta_{ik}$…
Pedro Gomes
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Is $L_p, loc(R)$ a Banach algebra?

I am not an expert in functional analysis, and I would like to know whether the space of locally integrabile functions, $L_{1,loc}(R)$ is a Banach algebra with respect to a suitable norm. The Wikipedia entry…
Guran Semiotovic
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Let $\{f_1,\dots,f_n\}$ be an orthonormal set. Prove that $\sum\limits_{I=1}^n |f_i(x)|^2\leq C^2$

Let $\mu(E)<\infty$. Let $C$ be a constant. A subspace $V\subset L^2(E)$ is defined such that $f\in V$ implies that $|f(x)|
user67803
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Trace class operators

I have a question concerning the definition of the square root of bounded linear operators. To introduce some notation: tr denotes the trace of linear operators and $\mathcal{L}(H)$ denotes the set of bounded linear operators, from H to H, where H…
David
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A doubt about projection operators in a Hilbert space

Suppose $H$ is a Hilbert space and $B(H)$ is the space of bounded linear operators on $H$. Let $\{E_{\alpha}\}_{\alpha\in A}$ be an arbitrary collection of orthogonal projection operators. Then let $E$ be the projection onto the smallest closed…
Landon Carter
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Proving that map is open

Let $X$ and $Y$ be Banach spaces, and $T\in B(X,Y)$ such operator that $imT$ has a finite codimension in $Y$ i.e. there exists vector space $V\subseteq Y$ that $dimV$ is finite and $Y=imT\oplus V$. Let us define Banach space $X_1=X\oplus V$ with…
XYZ
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Show that a subset $K$ of the Hilbert space $L_2[0,1]$ is closed.

Let $K$ be a subset of $L_2[0,1]$ consisting of the functions $f_n(x)=(1+2^{-n})e^{2\pi inx}$, where $n=1,2,3,\ldots$. Show that the subset $K$ of the Hilbert space $L_2[0,1]$ is closed. If I want show $K$ is closed , I have to show $f_n$ converges…
user45955
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Finding the "operatorial" norm of an operator

Let $T:\ell^2 \to \ell^2$ with $T(x_n)_n=(\dfrac{x_2}{2}, \dfrac{x_3}{3},...,\dfrac{x_n}{n},...)$ and $\ell^2 = \{(x_n)_n \in \mathbb{C} | \sum_{n=1}^{\infty} |x_n|^2 < \infty \}$ I already proved that $T$ is linear and bounded, i.e. $||T(x_n)_n||…
asd11
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How do you show in linear operator is bounded?

I am trying to work out an example my teacher gave in class and he skipped a few steps. We have $x= (x_1, x_2, x_3)$ Then we define a linear operator $T:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ by $Tx = (x_2+x_3, 2x_1, x_2)$. How do I work out that T…
spitfiredd
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Pointwise and Uniform Boundedness on Banach Spaces

I'm working on the following problem in Royden: As a consequence of the Baire Category Theorem we showed that a mapping that is the pointwise limit of a sequence of continuous mappings on a complete metric space must be continuous at some point.…
yoshi
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