Mathematics Stack Exchange
Mathematics Stack Exchange

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
2
votes
1 answer

Why can we assume the existence of these functions

This is 1.5.8 on page 15 of Dixmier's _C* Algebra_: Let $A$ be a C*-algebra. For each positive integer $n$, we have $A = A^n$. It is enough to show that each hermitian element $x$ of $A$ is a product of $n$ elements of $A$. Now, if $f_1, \cdots,…
ShinyaSakai
  • 7,846
2
votes
0 answers

Reflexive with Respect to a Norm

Let $E$ be a normed vector space and suppose that two norms $\| \, \|_1$ and $\| \, \|_2$ are equivalent. We are asked that if $E$ is reflexive with respect to $\| \, \|_1$, is $E$ also reflexive with respect to $\| \, \|_2$? What does it mean to…
Andy Tam
  • 3,367
2
votes
2 answers

Prove $\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}=\inf_n||A^n||^{\frac{1}{n}}$

Let $X$ be a normed space and $A\in B(X)$ (i.e.A is continuous linear map in $X$), prove that $$\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}=\inf_n||A^n||^{\frac{1}{n}}$$ It is easy to see that $||A^n||^{\frac{1}{n}}\leq ||A||$
89085731
  • 7,614
2
votes
1 answer

How to show linear operator $T:\ell_1\rightarrow \ell_1$ such that $\lVert T\rVert =1$ never attains norm.

I have a question concerning the following proof. Let linear operator $T:\ell_1\rightarrow\ell_1$ be defined as $$T(x_1,\ldots,x_i,\ldots)=((1-\frac{1}{1})x_1,\ldots,(1-\frac{1}{i})x_i,\ldots)$$ I have been able to show that $\lVert T\rVert=1$, but…
typewriter
  • 262
2
votes
2 answers

Weakly continuous function but not strongly continuous.

It is known that, if a function $f$ from a planar domain $D$ to a Banach space $A$ is weakly analytic [i.e. $l(f)$ is analytic for every $l$ in $A^*$], then $f$ is strongly analytic [i.e. $\lim_{h \to 0} h^{-1}[f(z+h)-f(z)]$ exists in norm for every…
Timon
  • 2,401
2
votes
3 answers

Functional Analysis, topology induced by infinite norm

In the $C[0,1]$ function space, do all norms induce a finer topology than the topology induced by infinite norm ($\sup$ norm)?
Maria
  • 239
2
votes
1 answer

Show that given any $x\in X$, there exists $(a_n)_{n=1}^\infty\subset \mathbb{C}$ such that $\sum_{n=1}^\infty a_ny_n=x$

Let $X$ be a separable infinite dimensional Banach space and let $\{y_n\}\subset X$ be a dense sequence in the unit ball of $X$. Show that given any $x\in X$, there exists $(a_n)_{n=1}^\infty\subset \mathbb{C}$ such that $\sum_{n=1}^\infty…
Extremal
  • 5,785
2
votes
1 answer

Coercivity of self-adjoint operator

Let $X$, $Y$ be real Hilbert spaces. Let $A\in\mathcal{L}(X,Y)$, let $A^{\ast}$ denote the Hilbert-adjoint, and let $I_X$, $I_Y$ denote the identity mappings on $X$, $Y$. Given the self-adjoint operator $B:X\times Y\to X\times Y$, \begin{equation*} …
User129599
  • 303
2
votes
0 answers

Minkowski Functional & Hahn-Banach

I'm trying to solve the following problem in Functional Analysis but I'm not sure how to do it. The only hint I got is that I will have to use the Minkowski Functional and Hahn-Banach Theorem to get this statement proved. But how? Can someone…
user372904
  • 315
2
votes
2 answers

Find the vector $h_0$ is $\mathcal{H}$ such that $L(h)=\langle h,h_0\rangle$

Let $\mathcal{H}=l^2(\mathbb{N}\cup \{0\})$, and $L:\mathcal{H}\to\mathbb{C}$ is defined by $L(\{\alpha_n\})=\sum_{n=0}^{\infty}{n\alpha_n}\lambda^{n-1}$, where $|\lambda|<1$. Find the vector $h_0$ in $\mathcal{H}$ such that $L(h)=\langle…
MathUser
  • 335
2
votes
1 answer

Why is the Heaviside step function locally integrable?

We define the Heaviside step function: \begin{equation} \mathcal{H}(x) = \mathbb{1} _{[0, \infty[}(x) \end{equation} Why is $\mathcal{H}$ an element of $L_{loc}^1(\Omega)$? I.e., $\forall K$ compact in $\Omega$, $\mathcal{H} \in L^1(\Omega)$? I've…
user242756
2
votes
1 answer

Proving operator is surjective

Let $T\colon \ell_1\to c_0^*$ be a operator, such that $$(Ta)(x)=\sum_{j=1}^\infty \alpha_j\xi_j$$ and $a=(\alpha_j)_{j=1}^\infty \in \ell_1$, $x=(\xi_j)_{j=1}^\infty\in c_0$. I want to prove that $T$ is surjective. Any ideas on how to approach…
user374257
  • 21
2
votes
1 answer

A metric space embedded into a Hilbert space

Suppose $X$ is a metric space but not a Hilbert space, is it possible that it can be isometrically embedded into a Hilbert space? Can anyone give an example?
89085731
  • 7,614
2
votes
1 answer

Show that there exist a constant $C$ s.t. $\|F\| \leq C\|f\|$

Let $X$ be a closed subspace of ${L^1 (0,2)}$. Suppose that for every $f \in L^1(0, 1)$ there exists an $F \in X $ whose restriction to $(0,1)$ is $f$. Show that there is a constant $C$ such that we can always choose an $F$ satisfying $\|F\| \leq…
user383517
  • 241
2
votes
1 answer

Weak-* convergence and trace-class operators

I am reading a proof by Barry Simon, and he makes a statement equivalent to the following: Let $X$ be the space of compact operators over a Hilbert space $H$. Let $\{y_n\}_n$, $y_n>0$, be a bounded sequence in the dual space $X'$ of trace-class…
Jas Ter
  • 1,539
1 2 3
…
99 100