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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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Operatornorm inequality

Let $$E:=(C^0([0,1]),\|\cdot\|_1) , \quad F:=(C^0([0,1]),\|\cdot\|_\infty),$$ with $$\|f\|_1:=\int_0^1\vert f(t)\vert \, dt,\quad \|f\|_\infty:=\sup_{t\in [0,1]}\vert f(t)\vert .$$ For $k \in C^0([0,1]\times[0,1])$ and $f\in E$ define…
Tobi92sr
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Functional analysis,$\ell^p$, continuous and uncontinuous operator

We observe $C[0,1]$ (the $\mathbb{C}$-vectorspace of all continous functions in $\mathbb{C}$ on $[0,1]$) as subspace of $L^p([0,1],\lambda)$ where $\lambda$ notes the Lebesgue-measure on $[0,1]$. Observe $c_{00}$ as subspace of $\ell^p$ for every…
Cornman
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Weak$*$ topology is metrizable iff $X$ is finite dimensional.

Let $X$ be a normed $\mathbb K$-linear space . Then the weak$*$ topology is metrizable iff $X$ is finite dimensional. We know that if X is finite dimensional then the weak topology on X is metrizable and also X is finite dimensional iff $X^*$ is…
Mini_me
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Let $E$ be a Banach Space. The set of all continuous linear transformations with inverse also continuous is open

Let $E$ be a Banach Space. Prove that the set of all continuous linear transformations with inverse also continuous is open in $\mathcal{L}(E,E)$ (the set of continuous transformations from $E$ to $E$). Consider the norm $\|T\| =…
user2345678
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Exponentials of operators

If $\{T_n\}$ converges in operator norm to $T$ does it follow that $\exp(T_n)$ converges to $\exp(T)$ at least in finite dimensions? I can handle this when the operators commute but not in general.
Kavi Rama Murthy
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Image of $A^\frac{1}{2}$ equal to image of $A$

Let $A\in B(H)$ be a closed range positive operator, and $A^\frac{1}{2}$ its square root. Clearly image of $A$ is a subset of image of $A^\frac{1}{2}~~$($R(A)\subset R(A^\frac{1}{2})$). How do we prove $R(A)= R(A^\frac{1}{2})$? I edite the question
niki
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converse of open mapping theorem in functional analysis.

Let $X,Y$ be two Banach spaces over $\mathbb K$ and let $T:X\to Y$ be $\mathbb K-$linear and continuous . Then how to show that 'if T is open then it is surjective?' Please someone help.. Thank you..
Mini_me
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2 answers

Separable version of Banach Alaoglu

I have a question regarding the following statement: Let $X$ be a separable Banach space, then every bounded sequence in $ X^\ast $ has a weak$^\ast$-convergent subsequence. I am sure that it suffices $X$ to be a normed vector space. However in most…
user317721
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1 answer

$U(x,r)\cap U(y,\frac{1}{n})\neq \emptyset $.

I have to prove that If $X$ is a normed space and $y \in B(x,r) $ (where $x \in X$ and $r>0$) then $U(x,r)\cap U(y,\frac{1}{n})\neq \emptyset $. I have tried to prove playing with the norms but I have not been able to. P.S: $B(x,r)$ is a close ball…
Juan ALmeida
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Is the differentiation operator an open mapping?

Consider the vector spaces $C([0, 1])$ and $C^1([0, 1])$ with norm, \begin{align*} \displaystyle \|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|, \end{align*} and let $T:C^1([0, 1])\to C([0, 1])$ the operator given by, \begin{align*} \displaystyle…
PtF
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How to show $||T||= \sup\{{||T(x)|| \over ||x||} : x \in X , x \ne 0\}$?

Let $X,Y$ be normed $\mathbb K$ linear space . $X \ne \{0\}$ and let $T$ be a $\mathbb K$ linear function. Then $$\|T\|= \sup\left\{\frac{\|T(x)\|}{\|x\|} : x \in X , x \ne 0 \right\}.$$ Here $\|T\| = \sup \{\|T(x)\|: x \in X, \|x\| \leq 1 \}$ How…
MADmind
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Kernel of the differential operator

I've read somewhere that the kernel of a linear map is closed iff the map is bounded. Consider the derivative operator $D: \mathcal C^1([0,1],\mathbb C) \to \mathcal C([0,1],\mathbb C)$, i.e. $Df = f'$. Since this map is not bounded, the kernel must…
Joe G.
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Showing that $r(T)\leq \omega(T)$?

Let $E$ be a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, with inner product $\langle\cdot\;| \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{L}(E)$ the algebra of all bounded linear operators from $E$ to $E$. Let $T\in…
Student
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About weakly continuous functions

Let $C_{w} ( [0,M], H)$ be the space of weakly continuous functions from $[0,M]$ into Hilbert space $H$. Then does " $u \in C_w( [0,M], H)$ " mean $$ \text{for any}\; \{ x_n \} \subset [0,M] \;\text{satisfying} \; \lim_{n \to \infty} x_n = x, \\…
Ann
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Real symmetric operators in $\ell^p$, for $p\neq 2$.

Consider the spaces $\ell^p$, for $1 \leq p \leq \infty$. Suppose we have a bounded operator $S: \ell^\infty \to \ell^\infty$ such that $S(\ell^p) \subseteq \ell^p$ for every $p \geq 1$ and such that $S$ restricted to $\ell^p$ is also a bounded…
Ruben
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