[Intro to Topology] Verify if $K$ is compact
Consider $C[0, 1]$ with the norm $||f||$ =$\int_{0}^{1}|f(x)|dx$. Verify if the set $K = \{f ∈ C[0, 1] : f(0) = 0 = f(1) and ||f|| = 1\}$ is compact.
We were just introduced to compact metric spaces, so we don't have much beyond the definitions by open coverages and sequeneces.
Answer
Answers can be viewed only if
- The questioner was satisfied and accepted the answer, or
- The answer was disputed, but the judge evaluated it as 100% correct.
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to 50% commission on every question your affiliated users ask or answer.
- answered
- 191 views
- $4.99
Related Questions
- Convex subset
- Prove that a closed subset of a compact set is compact.
- Pathwise connected
- Knot Theory, 3-colourbility of knots
- Undergrad algebraic topology proof
- Let $(X, ||\cdot||)$ be a normed space. Let $\{x_n\}$ and $\{y_n\}$ be two Cauchy sequences in X. Show that the seqience Show that the sequence $λ_n = ||x_n − y_n|| $ converges.
- Prove that $S \subseteq X$ is nowhere dense iff $X-\overline{S}$ is dense.
- Banach's fixed point theorem application
