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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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On the necessity of being a *dense* subset in completion of a metric space

Quoted from the book Introductory functional Analysis by Erwin Kreyszig : 1.6-2 Theorem (Completion). For a metric space $X = (X, d)$ there exists a complete metric space $\bar{X}=(\bar{X}, \bar{d})$ which has a subspace $W$ that is …
user231343
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Is the following subset compact?

Let $E$ be a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For $M\in \mathcal{L}(E)^+$, we consider \begin{eqnarray*} B_M &=&\{x \in E;\;\;\;\langle Mx\;,\;x\rangle\leq1\}. \end{eqnarray*} Is …
Student
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$L^{2}(\Omega)$ estimate of a exponential.

I need to prove that: if $\left\| u \right\|_{H^{1}(\Omega)} \leq 1$, exists $\sigma$, K such that $$\left\| e^{\sigma |u(.)|^{2}} \right\|_{L^{2}(\Omega)} \leq K.$$ I've tried so far: If $u$ has this limitation, then $\left\| u…
Lucas Galhego
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Operators in Hilbert space ( orthogonal projection)

Some one help me to solve this problem i put some hints Show that every projection $P \in \mathcal{B}(H)$ is and extreme point in the convex set $$B_+ = \{T \in \mathcal{B}(H) : T \geq 0, \Vert T\Vert \leq 1\}$$ Hints: If $P = \lambda S + ( 1 –…
math
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Why the following subset is convex??

Let $M$ be a positive matrix on $E$. Consider the following subset $$ W_M(T)=\left\{\lambda\in \mathbb{K};\;\exists x \in E;\,\|x\|_M^2:=\langle Mx,x\rangle=1 \;\hbox{and}\;\langle M T x,x\rangle=\lambda\right\}.$$ Why $W_M(T)$ is convex? Thanks!
Student
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How to prove that there is an the embedding of $H$ onto $H^*$

I have defined the map $\alpha:H\to H^*$ given by $$(\alpha x)(y) = \langle y, x\rangle.$$ I want to use it to show that: (1) it is an isometric imbedding of $H$ onto $H^*$, (2) and that $\alpha(\lambda x + \mu z) =\bar{\lambda} \alpha x + …
Sulayman
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Banach space and bounded below operators

Let $T\in L(X,Y)$, where $X, Y$ are Banach spaces. How can I conclude from $\ker(T) = \{0\}$ and $\mathrm{im}(T)$ closed that $T$ is bounded below, e.g. there exists a $c>0$ such that $||Tx||\geq c ||x||$?
samsons
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Continuous linear functional on $\mathbb{C}^A$ with the product topology

My homework: Let $A$ be a set. Endow $\mathbb{C}^A$ with the product topology. Then any continuous linear functional $\Lambda:\mathbb{C}^A\rightarrow\mathbb{C}$ is of the form $\Lambda(f)=\sum_{i=1}^n \alpha_i f(a_i)$ for some complex…
Gils
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Dirichlet and Neumann boundary conditions Finite Element Method

I have the following problem in Finite Element Method $$ -(\alpha u')' + \beta u' + \gamma u$$ with $ \Omega = (0, 1)$, $ u(0) = 0 $ and $ u'(1) = 3 $ to be able to write the weak formulation of the problem do I need a lifting function? i.e., do I…
BRabbit27
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completeness of a functional space

The function space $C([a,b])$ is the space of all continuous, real valued functions defined on $[a,b]$ and has the supremum norm ($\|\cdot\|_{\infty}$) associated with it. I came across a book which says that the supremum norm is complete on this…
InMyOwn Euclidean space
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Why a linear operator is bounded if $|f(x)|\leqslant C\|x\|$

It's written in my book that $L:(X,|\cdot |_X)\to (Y,|\cdot |_y)$ is bounded if there is $C>0$ s.t. $$|L(x)|_Y\leqslant C|x|_X.$$ This definition seems very strange to me. Where is the motivation behind this definition ?
user330587
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Regularized Urysohn function

I have been told to look for a regularized version of Urysohn functions, that is, a function which is non-zero in an open open set $U \subset \mathbb{R}$, is one restricted to a compact set $K$ contained in $U$, is zero outsdie $U$; and such that it…
busman
  • 781
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find $M^{\perp}$ in Hilbert space

Given $n \in \mathbb{Z}^+$ and $M = \{ (x_1,x_2,..,x_n,0,0,...) \mid x_1,x_2,..,x_n \in \mathbb{R} \}$. Find $M^{\perp}$ in $l^2$? I can show $M$ is a closed subspace of $l^2$ and a Hilbert space. Let $P_M : l^2 \to M$: $$x= (x_1,x_2,..,x_n,..)…
Desunkid
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closed operator in Hilbert spaces

If $X$ be a closed subspace of $L^2([0,1])$ and each element in $X$ belongs to $L^{\infty}([0,1])$, So $X$ is closed in $L^{\infty}([0,1])$ and $\lVert f \rVert_2 \leq \lVert f \rVert_{\infty} $. Therefore X is Banach space. If I define a map $T: X…
Tien
  • 139
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How to show closed unit ball is compact?

I know that in normed linear space closed unit ball is compact iff space is finite dimensional. (I proved this by using Riesz lemma). My question is how to prove same statement for inner product space ( without using Riesz lemma)
CHOUDHARY bhim sen
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