# Tagged Questions

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### Every Bounded set contained in a Compact set

In a general metric space, is every bounded set contained in a compact set?
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### Example of a sequence of functions

Construct an example of a sequence of functions $(f_n)$ defined on $[0,1]$ such that $f_n$ converges pointwise to $0$ and for every sequence of numbers $(a_n)$ that tends to $\infty$, sequence ...
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### Some properties about $L^p$ with $0<p<1$

We are coming across many Banach spaces $L^p$ with $1\leq p\leq\infty$. But how about $0<p<1$? Can it be normed? How about its metric induced by the norm? And how about its ...
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### Interior of a convex set is convex [duplicate]

A set $S$ in $\mathbb{R}^n$ is convex if for every pair of points $x,y$ in $S$ and every real $\theta$ where $0 < \theta < 1$, we have $\theta x + (1- \theta) y \in S$. I'm trying to show that ...
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### Bounded Semivariation and Bounded Variation in locally convex TVS

Let $(X,\tau)$ be a Hausdorff locally convex TVS and let $P(X)$ be a family of seminorms on $X$ that generates $\tau$. We consider the following definitions. Definition 1. A function $f:[a,b]\to X$ ...
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### show $\langle f,g \rangle _w = \int^b_a f(x)g(x)dx$ is an inner product

Let $w(x)$ be a strictly positive continuous function on [a,b]. Define a form on $C[a,b]$ by the formula $\langle f,g \rangle _w = \int^b_a f(x)g(x)dx$ for $f,g \in C[a,b]$. Show that it is an inner ...
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### Show that the norm of the derivitive of a $C^1$ function over a vector space is non-negative, homogeneous and satisfies the triangle ineq

For $f$ in $C^1[a,b]$, define $p(f)= \parallel f'\parallel _{\infty}$. Show that $p$ is non-negative, homogeneous, and satisfies the triangle inequality. Why is it not a norm? -I can easily show the ...
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### Problem 5. ( chap3. p.87, functional analysis, W.Rudin)

I had done part a, b, and d,. But i cannot breakthrough part c, and part e,. I restate entired problem in the following: For $0<p<\infty$, let $l^p$ be the space of all functions $x$ (real or ...
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### About the continuity of $B$ (problem 12 chap.2, p.55, functional analysis, W.Rudin)

Let $X$ be the normed space of all real polynomails in one variable, with $||f||=\int_0^1 |f(t)|dt$. Put $B(f,g)=\int_0^1 f(t)g(t)dt$, and show that $B$ is a bilinear functional on $X\times X$ which ...
### What is the appropriate topology on $C_c^\infty (\mathbb{R}^d)$?
Let $\{ U_k:k\in \mathbb{N}\}$ be an increasing sequence of open subsets of $\mathbb{R}^d$ whose union is $\mathbb{R}^d$ and such that each $K_k:=\overline{U_k}$ is compact and $K_k\subseteq U_{k+1}$. ...