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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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A bounded linear operator between Banach spaces that cannot be compact

Let $K: X \to Y$ be a bounded linear operator, where $X$ and $Y$ are two Banach spaces. Further assume that the image $imK$ is a $\infty$-dim closed subspace of Y. In my script they claim that in such a setting $K$ can never be compact because by…
noctusraid
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Show that $\Vert T\Vert = \sup\{\vert \langle Tx, y\rangle\vert : x,y\in E, \,\Vert x\Vert \leq 1, \,\Vert y\Vert \leq 1\}$

I have the following task where I'm blocked: Show that if $E$ is an inner product space and $T: E\rightarrow E$ is a bounded linear operator, that is $$\underset{\Vert x\Vert \leq 1}{\sup} \Vert Tx\Vert < \infty,\quad x\in E$$ then $$\Vert…
jjepsuomi
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Hahn Banach extension of a given linear functional

Consider $\mathbb R^2$ with $||.||_1$ norm and $M=\{(x,0):x\in \mathbb R \}$. Define $g:M \to \mathbb R$ by $g(x,0)=x$. Then which one is the Hahn-Banach extension $f$ of $g$? $f(x,y)=2x $ $f(x,y)=x-2y$ $f(x,y)=x+2y$ $f(x,y)=x+y$ According to…
user303866
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Proving a complex version of the Hahn-Banach theorem

Consider a subspace $M$ of a complex normed vector space $(X,\|\cdot\|)$, $p:X\to\mathbb{R}$ sublinear and $f:M\to\mathbb{C}$ linear such that $|f(x)|\le p(x)$ $\forall x\in M$. We want to prove that there exists $\Lambda:X\to\mathbb{C}$ such that…
Jason Born
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topology of normed spaces

Following is a list of problems from an exam for admission into Ph. D program. I have just compiled all previous questions on topological properties of certain subsets of normed linear spaces and i tried to work out . I would be thankful if some one…
Abhishek Shrivastava
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$U^\dagger U=id \nRightarrow UU^\dagger=id$ for infinite dimensional vector spaces

I wasn't sure whether this question was appropriate for math.stackexchange or physics.stackexchange. I don't really have an understanding of bounded linear operators or Hilbert spaces so this question may be slightly informal. Nevertheless, let $U$…
Muster Mark
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How to calculate this Frechet derivative

Suppose $F:C^1(\Omega, [0,T]) \to C^1(\Omega, [0,T])$ with $$F(u) = u_t - f(x, t, u, u_x).$$ How do I calculate the Frechet derivative of $F$ at the point $w = f(x,t, 0, 0)t$? It should be $$F'(w, v) = v_t - \frac{\partial f}{\partial z}\bigg|_{w}v…
Court
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Why does the weak topology make the functions in $E'$ continuous?

Let $E$ be a Hausdorff locally convex topological vectorspace. Consider the source $p_f\colon E\to\mathbb{R}$, where $p_f(x)=\lvert f(x)\rvert$ and $f\in E'$, the continuous dual of $E$. The maps $p_f$ are semi-norms, so they generate an initial…
Jeroen
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Significance of the Mazur-Ulam Theorem

The Mazur-Ulam theoem states that any surjective isometry between any two real normed spaces is necessarily affine. What is the significance of the theorem? It is said that the birth of the Banach space theory is due to this theorem[1], but I don't…
Idonknow
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Laplace Operator of Norm is Dirac Distribution?

I want to show that: $\Delta f = c \delta_0$ where $f(x) = \frac{1}{||x||_2^a}$ and $\delta_0$ is the $n$-dimensional Dirac distribution. $n$ is a natural number and I want to find $a(n)$ so that the above equation is true. I tried calculating out…
Emil
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Is it true $\left \| T(x) \right \| \le \left \| T \right \| \cdot \left \| x \right \| $

Let be $E$ a real Banach space and $E^*$ the real dual space. $\forall T \in E^*$ Is it true $\left \| T(x) \right \| \le \left \| T \right \| \cdot \left \| x \right \| $? Why? Let be $\{ T_n \}_n$ a Cauchy's sequence. Is it true $\left \|…
Skills
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How do I find a bounded linear functional under the assumptions of the following theorem?

Theorem: Let $X$ be a normed space and $0 \not= x_0 \in X$ be a arbitrary. Then there exists a bounded linear functional $\bar f$ on $X$ such that $$\|\bar f \|=1, \quad \bar f(x_0) =\|x_0\|.$$ Problem: Find $\bar f$ when $X$ is the Euclidean…
Mark
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Orthogonal projection - Brezis Functional Analysis Exercise 5.8

I'm trying to solve one exercise from Brezis "Let $A$ be a measure space and let $h : A \rightarrow [0, +\infty)$ be a measurable function. Let $K=\{u ∈ L^2(A); \lvert u(x)\rvert \le h(x)\ \text{a.e. on} \ A\}$. Check that $K$ is a nonempty closed…
Willian Nunes
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Applications of the Hahn Banach theorem for normed spaces?

If $X$ is the Euclidean space $\mathbb{R}^3$ and $f(x)=a_1 x_1 + a_2 x_2$, $x=(x_1, x_2) $ a bounded linear functional on the subspace $\mathbb{R}^2$ of $X$, how do I find a bounded linear functional $f'$ that is an extension of $f$ with the same…
Umer Jadoon
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If $X$ is reflexive, $T\in B(X,Y)$ and $\| Tx_n - Tx \| \rightarrow 0$ whenever $x_n \rightarrow x$ weakly then $T$ is compact.

I want to prove: If $X$ is reflexive, $T\in B(X,Y)$ and $\| Tx_n - Tx \| \rightarrow 0$ whenever $x_n \rightarrow x$ weakly then $T$ is compact. Here is my attempt at the proof: Assuming all the conditions hold, we'll want to show $T(ball X )$ has…
Euthenia
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