Bounded and Continuous Linear Transformations

In this video definitions bounded and continuous functions are given on a normed linear space.
Further, the following Theorem is given
Theorem:Let T : N to N' be a linear transformation from a normed linear space N to normed linear space N'. then the following statements are equivalent:
1. T is continuous
2. T is continuous at the origin
3. T is bounded
4. If S is a sunset of N containing all those elements whose norm are less than 1, then T(S) will be a bounded subset of N'.

[Rakeshkumar-jfty]

Excellent sir... Aisa lag raha hai jaise offline hi padh rahe hai bus thoda aapka hasi mazak wala punchline miss kar rahe hai... 🤣

[Rakeshkumar-jfty]

24:35 wala term || T(yn) ||>1
Or || limT(yn) ||>=1 samjh nhi aaya sir thoda sa help kare sir...

[jayachauhan]

let x and y be normed spaces over the field k and let B (x, y) be the lnear spaces of all bounded linear operators
T: X -- Y . Define ||.|| :B (X, Y) -- R by ||T || = sup{ ||Tx|| }y : xex, ||x|| x <1}
then (B (X, Y, ) |||.||) is a normal space . Further more, if Y is a banach space, then B( X, Y ) is a banach space.

[teendairy]

Introduction change krna chahie apko ye bilkul annoying h