# 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).

31,121 questions
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### does the definition of continuity require that the domain is the reals?

When we are talking about continuity at $c$. We say for a given epsilon, there is a distance delta such that for all $x$ within this distance of $c$, $|f(x)-f(c)|<\epsilon$. What if there are some ...
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### Difference of Closed Convex sets in Banach Space

Let $A$ be a closed, convex, set in a Banach space $X$, and let $B$ be a closed, bounded, convex set in $X$. Assume that $A \cap B = \emptyset$. Set $C = A- B$. Prove that $C$ is closed, and convex. ...
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### Duality pairing in Banach spaces

Suppose we have a Banach space $X$ and its dual $X^{'}$. Then the duality pairing is often written $$\langle f,v\rangle_{X^{'},X} = f(v)$$ for $f \in X^{'}$ and $v \in X$. But is it allowed to ...
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### Projection from point onto plane

Let the plane is $\prod:\vec{x}\bullet\hat{n}=d$ be a plane, where $\vec{x}\in R^{3}$ and $d\in R$, then can I show using norm and Cauchy-Schwarz inequality that projection of point $x$ onto the ...
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### find extrema min,max on a multivariable function

I have to evaluate min,max inside and on the boundary of a domain: $D=\{xy-1\le 0,|y-x|\le1\}$ $f(x,y)=(y-x)e^{xy}$ So That's a ($xy-1$) hyperbola and two lines. I proceeded like so: for $y=1+x$ ...
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### Let $\alpha >1.$ Then $\forall x\gt 0: \psi(\alpha x)\leq \alpha \psi( x)\;.$ True or False?

Let $\psi$ be a function satisfying : $\psi: \mathbb{R}^+\rightarrow \mathbb{R}^+$ . $\psi$ is non-decreasing. $\psi (x)< x, \forall x> 0$. I want to know if the following statement is ...
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### If $X$ is a real normed linear space and $r>0$, then $B_{r}(x_0+y_0)=B_{r}(y_0)+\{x_0\}$ for fixed $x_0,\,y_0\in X.$

Let $X$ be a real normed linear space and for $r>0$, let $$B_{r}(x_0)=\{x\in X:\|x-x_0\|\leq r\}.$$ I want to prove that $B_{r}(x_0+y_0)=B_{r}(y_0)+\{x_0\}$ for fixed $x_0,\,y_0\in X.$ My attempt ...
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### How is the dual cone of a subspace its orthogonal complement?

From Boyd and Vandenberghe's Convex Optimization: A dual cone of a subspace $V \subseteq \Bbb R^n$ is it's orthogonal complement. $V^{*} = \{y : v^Ty = 0, \forall v \in V\}$ but the dual ...
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### Compact open operator between Banach spaces

Let $X,Y$ be Banach space, $Y$ infinite dimensional. Show that no $T \in \mathcal{K}(X,Y)$ is open. By definition $T$ is open if and only if $\exists r >0$ such that $B_Y(0,r) \subset T(B_X(0,1))$ ...
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### Prove or Disprove: Summation of two functions (at least one discontinuous) supports IVP, if both of them support IVP.

Let, f and g be two functions on R, support IVP and at least one them is discontinuous. Then prove or disprove ( with example) whether f+g also supports IVP. If f and g, both are continuous, then it ...
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### Prove that if $F: X \longrightarrow C(Y, Z)$ is continuous, then $f : X \times Y \longrightarrow Z$ is continuous. [on hold]

The space $C(Y, Z)$ is supposed to be a topological space with topology of uniform convergence. I wanted to prove the fact using the definition of continuity, but I've failed. I really don't ...
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### Rudin's functional analysis appendix A4 (a)

Quick question about the following theorem: If $K$ is a closed subset of a complete metric space $X$ then the following three properties are equivalent: (a) $K$ is compact (b) Every ...
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### Compact operator on $L_2([0,1],m)$

Consider the Hilbert space $H=L_2([0,1],m)$ where $m$ is the Lebesgue measure on the interval $[0,1]$. Let $T \in \mathcal{L}(H,H)$ given by \begin{equation*} T\ f(x)=x \ f(x) \ \ \ \ f \in H,\ x \...
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### Compactness in weak$^*$-topology on $l_1(\mathbb{N})$

Let $K$ denote the closed unit ball of $l_1(\mathbb{N})$ (considered as a vector space over $\mathbb{C}$). Is $\mathrm{co}(\mathrm{Ext}(K))$ (the convex hull of its extreme points) compact in the ...
A two player game. Players choose a number from a segment $[0;1]$. A payoff function of the first player $f_1(x,y)=|x-y|,$ where $x$ and $y$ - the numbers chosen by the first and the second players ...