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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 Bochner space

What does $u_n$ converges weakly to $u$ in $H = L^2(0,T,H^1_0)$ mean explicitly? My thinking is; the dual space of $H$ is $H^* = L^2(0,T,H^{-1})$. So this means for every $f \in H^*$, $f(u_n) \to f(u)$. How to write this out in inner products?…
soup
  • 1,414
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Bilinear operator norm question.

So I wrote this down a week ago and cannot figure out what I was thinking. Not sure if this is correct. Context we have a bilinear operator $B:X\times Y\to \mathbb{K}$. Is it true that $$\sup_{x\in X, y\in Y} \|B(x,y)\|< \infty \implies…
Matthew
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If every $x\in X$ is uniquely $x=y+z$ then $\|z\|+\|y\|\leq C\|x\|$

Problem Statement Let $(X,\|\cdot \|)$ be a Banach space and $Y$, $Z$ closed subspaces of $X$. If every $x \in X$ can be uniquely represented as $x=y+z$ for $y \in Y$ and $z \in Z$ then show that there exists $c$ such that $\|y\|\leq C\|x\|$ and…
user210552
  • 807
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functional analysis complementary subspace

Let $Y$ and $Z$ be closed subspaces in a Banach space $X$. Show that each $x \in X$ has a unique decomposition $x = y + z$, $y\in Y$, $z\in Z$ iff $Y + Z = X$ and $Y\cap Z = \{0\}$. Show in this case that there is a constant $\alpha>0$ such that…
math
  • 305
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Show that the norm of a complex functional is equal to the norm of its "real part"

The statement of the exercise is pretty simple to understand, but I'm having trouble to prove it. The exercise is the following: (Exercise 3, from Introduction to functional analysis, by A. E. Taylor, page 190).: If $X$ is a complex normed linear…
Mancala
  • 796
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Find counterexample product of weak$^*$ convergent product

The following assertion is true in the case $1
tubmaster
  • 708
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Weak star null sequence of norm one vectors in infinite dimensional Banach space

Suppose that $(X, \|.\|)$ is an infinite dimensional Banach space. I would like to ask whether we could construct a sequence $\{x_n^*\}_{n\in \mathbb{N}}\subset X^*$ (dual space of $X$) such that: $\|x_n^*\|_{X^*}=1$; $\{x_n\}$ is weakly convergent…
blindman
  • 3,117
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Functional equation : prove a property of the solution

Let the equation be $f (a,b)=\int_\Omega g(a,x)g(b,x)dx $ If the condition on f is : $f (a,b)=h (a-b) $ can we prove that $g (a,x) $ is of the form $k (a-x) $ ?
QuantumPotatoïd
  • 286
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1 answer

Inverse of $I - T$ where $\lVert T \rVert < 1$

Let $T$ be a bounded linear operator on $E$ with norm strictly less than 1. Do we need $E$ to be Banach to define $(I - T)^{-1}$ as $$ \sum_{k=0}^\infty T^k ?$$ The proof I am aware of for $(I - T)^{-1} = \sum_{k=0}^\infty T^k$ pointwise requires…
user369210
  • 1,339
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2 answers

Is Wirtinger's inequality valid on the space if $f$ is non-zero on the boundary?

we know that $\pi^2 \int_0^a |f|^2 dx \leq a^2 \int_0^a |f'|^2 dx$ if $f$ is $C^1$ and $f(0)=f(a)=0$. I am interested in is this inequality also valid if $f(0) \neq 0$?
Huili Zhang
  • 21
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$T \in B(X,Y)$ Hilbert Spaces: $ \|\ T \|\ = \sup_{\|\ x \|\ = 1 = \|\ y \|\ } | \langle y , T(x) \rangle | $

Question : For $T \in B(X,Y)$, where $X$ and $Y$ are Hilbert Spaces, $$\|\ x \|\ = \sup_{\|\ y \|\ = 1 } | \langle y , x \rangle | \hspace{1cm} \text{and} \hspace{1cm} \|\ T \|\ = \sup_{\|\ x \|\ = 1 = \|\ y \|\ } | \langle y , T(x) \rangle |…
Dragonite
  • 2,388
2
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2 answers

Norm of a linear functional

Consider $C[0,1]$ (the space of continuous functions on $[0,1]$) with the max-norm (assume the underlying field is $\mathbb{R}$). For $g \in C[0,1]$, define $\Phi_g: C[0,1] \rightarrow \mathbb{R}$ by \begin{equation*} \Phi_g(f) = \int_0^1 f(t)g(t)…
ragrigg
  • 1,675
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1 answer

$\phi : L^1(A) \to \mathbb{C}$ where $f \mapsto \int gf$ for $g \in L^{\infty}(A)$ for $A \subseteq \mathbb{R}$ compact isometric isomorphism

Full Question: The map $L^{1}(A) \to \mathbb{C}$, $f \mapsto \int gf$ is linear, and continuous when $g \in L^{\infty}(A)$. Assuming surjectivity, show $L^{1}(K)^*$ is isometrically isomorphic to $L^{\infty}(K)$ for $K \subseteq \mathbb{R}$ is…
Dragonite
  • 2,388
2
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1 answer

Unit ball of Banach space remain closed in embedding

Let $Y\subset X $ and $X $ and $Y $ be Banach spaces with $$ \|y\|_X \leq C \|y\|_Y $$ for all $ y\in Y $. Show that the unit ball of $Y $ is closed in $X $. In other words, I want to show that $Y $ semi-embeds into $X $. I am sure this is…
Matt
  • 2,187
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Euler's Method and induction

Euler's Method for series associates with a given series $\sum\limits_{j=0}^\infty(-1)^ja_j$ the transformed series $\sum\limits_{n=0}^\infty\frac{\Delta^n a_0}{2^{n+1}}$ where $\Delta^0a_j=a_j$, $\Delta^na_j=\Delta^{n-1}a_j-\Delta^{n-1}a_{j+1}$,…
Burgundy
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