David Spade Comfort from a Friend After Kate Spade Suicide

6/5/2018 3:07 PM PDT

David Spade Looking Sad in L.A. Shortly After Kate Spade Suicide News

6:25 PM PT -- David shared a photo of Kate, whom he calls "Katy," at his book signing ... along with a heartfelt and encouraging message.

David Spade leaned on a friend hours after getting the tragic news about sister-in-law Kate Spade's death.

The actor and comedian was out in L.A. Tuesday afternoon, looking solemn as he left home. His brother, Andy, was married to Kate and ... as we reported, he was home when she hanged herself.

David wore sunglasses and kept his head down, but was seen getting a consoling hug from a friend.

Kate and David's bro launched Kate Spade Handbags together in 1993 and got married in 1994. They have a daughter together -- 13-year-old Frances Beatrix.

We broke the story ... Kate left a suicide note behind addressed directly to Frances that said, "Bea - I have always loved you. This is not your fault. Ask Daddy!"

Kate was 55.