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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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If $T_nx\to Tx$ and $x_n\to x$ then $T_nx_n\to Tx$

Let $X,Y$ be Banach spaces, $T_n:X\to Y$ linear and bounded with $\lim_{n\to\infty}T_nx=Tx$ for every $x\in X$. If $x_n\to x$ in $X$ then we have $T_nx_n\to Tx$ in $Y$. What I'm trying to do is to check for triangle inequalities…
EliotTimmy
  • 87
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Show that $||x-y|| > 1/2$ for all $y \in Y$.

I'm sorry for posting this question, but I really have no clue on how to proceed: Let $E$ be a normed vector space. If $Y$ is a closed proper subspace of $E$, then there is a $x \in E$ such that $||x|| = 1$ and $||x - y|| > 1/2$ for all $y \in…
Andy Tam
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Prove that if a functional is in the dual space then the null space of the functional is closed.

$X$ is a normed vector space and $T \in X^*$. Show that \begin{equation} \label{prob} T \in X' \Leftrightarrow N(T) \; \text{is closed}, \end{equation} where $N(f)$ is the null space of $T$. i) Prove that if $T \in X'$, then $N(T)$ is…
ANYN11
  • 365
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Compute operator norm of all absolutely continuous fuctions

Let $H=$ the collection of all absolutely continuous functions, $ f:[0,1]\to \mathbb{C}$, such that $f(0)=0$ and $f'\in L^2(0,1)$. If $=\int^1_0f'(t)\overline{g'(t)}$ for $f$ and $g$ in $H$, fix $t$,$0
noname1014
  • 2,513
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Let X be a linear space, why is the dual space of X a Banach space?

Let X be a linear space, why is the dual space of X a Banach space? Can anyone please explain?
user1559897
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sum of elements converge in Hilbert space implies convergence of square

I have a question about a statement which I don't know if it is true. Given a Hilbert space $H$, if we have a sequence of element $\{c_k\}$, with $||\sum_I c_k||\leq A$, where $I$ is any finite index set and $A$ is independent of $I$. Can we say…
Ale
  • 41
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Decomposition into Simple Elements

I'm Nina. I have a really tough homework that counts as a test and I couldn't do it. It's really urgent. Help please! Let $\alpha \in \Bbb{R}^+$. Let $F$ be the function defined in $\Bbb{R}$ by $$F(x) = \frac{1}{\cosh\alpha - \cos x}.$$ What is the…
Nina
  • 21
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assume $X$ is a normed space with separable dual $X'$. Show that $X$ is a separable space.

assume $X$ is a normed space with separable dual $X'$. Show that $X$ is a separable space. What have i proven already: 1) If $C$ is a countable subset of $X$, such that $\overline{Sp}(C) = X$, then $X$ is separable. 2) If $A$ is a subset of $X,…
Kees Til
  • 1,958
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A quotient space is isomorphic if we have closedness

Let $\mathcal{X}$ and $\mathcal{Y}$ be Banach spaces, $T\in L(\mathcal{X},\mathcal{Y})$, $\mathcal{N} =\{x: Tx = 0\}$, and $M = range(T)$. Then $\mathcal{X}/\mathcal{N}(T)$ is isomorphic to $\mathcal{M}$ iff $\mathcal{M}$ is closed. I know that by…
Wolfy
  • 6,495
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How to show $A$ is a bounded linear operator?

If $H$ is a Hilbert space and $A:H \rightarrow H$ is a linear operator such that $\langle Ax, y\rangle = \langle x, Ay\rangle, \forall x, y \in H$ then how to show $A$ is a bounded linear operator?
thomus
  • 577
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The interior of $C^1([0,1])$ is empty

The interior of $C^1([0,1])$ w.r.t. $\Vert \cdot \Vert_{\infty}$ is empty My intuition is that we have to use the fact that $C([0,1])$ is dense in $C^1([0,1])$. But I don't really know how to formulate a solution formally. My solution: Assume it…
MorganeMaPh
  • 507
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1 answer

Comparison of norms

Considering the comparison of norms I have the following proposition from a certain book: Let $X$ be a vector space and let $\Vert \cdot\Vert_{1}$ and $\Vert \cdot\Vert_{2}$ be two norms on this space. $\Vert \cdot\Vert_{1}$ is weaker than $\Vert…
Nirav
  • 1,202
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Proof using Banach fixed point theorem

Theorem.$\ $ Let $E$, $F$ be two Banach spaces, $U$ an open ball in $E$, $V$ an open ball in $F$ of center $y_0$ and radius $\beta$, and $v$ a continuous mapping of $U \times V$ into $F$. Suppose that there exists a constant $k$ with $0 < k < 1$…
Randy Randerson
  • 866
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Completion of a normed space using an isometry

I'm practicing some old exams for my functional analysis exam tomorrow, and i'm having trouble with the following: Let $X$ be a reflexive Banach space and let $Y$ be a normed space. Assume there exists a linear map $T: Y\to X$ which is an isometry.…
ronalddb89
  • 141
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Relation between Ranges of compact operators

I am reviewing functional analysis and getting stuck in this problem. Let $X,Y$ be two Banach spaces and $A,B\in L(X,Y)$. Prove that if $A$ is a compact operator and $R(B)\subset R(A)$ then $B$ is also a compact operator. Can anyone give me some…
Omega
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